Christians claim the resurrection story to be very unique. Is there any story in Puranas , where anyone died and resurrected after dying. And also is there is any parallel story of someone raising people from the dead.
There is One intersting story in Valmiki Ramayna (Yuddha Kanda - Book Of War) about Resurrection of the Monkeys by Lord Indra , who were killed in the battle between Shree Rama and Ravana.
समुत्तिष्ठन्तु ते सर्वे हता ये युधि राक्षसैः | ऋक्षाश्च सह गोपुच्छैर्निकृत्ताननबाहवः || ६-१२०-१३
- te sarve = (Let) all those monkeys; R^ikShaashcha = and bears; gopuchchhaiH = along with long-tailed monkeys;ye hataaH = who have been killed; yudhi = in battle;raakShasaiH = by the demons; nikR^ittanana baahavaH = and whose heads and arms have been severed;samuttiShThantu = be active again.
"Let all those monkeys and bears along with long-tailed monkeys, who have been killed in battle, by the demons and whose heads and arms have been severed, be alive and active again."
मत्प्रियेष्वभिरक्ताश्च न मृत्युं गणयन्ति ये | त्वत्प्रसादात्समेयुस्ते वरमेतमहं वृणे || ६-१२०-८
- te = those monkeys; ye = who; na gaNayanti = did not take into account; mR^ityum = their death; abhiraktaaH matpriyeShu = and who were fond of doing favour to me; te sameyuH = let them get reunited; (with their near and dear ones); tvatprasaadaat = by your grace; aham = I; vR^iNe = seek; etam = this varam = boon (of you).
"Those mokeys - who coutned death as nothing and who were fond of doing favour to me - let them get re-united with their near and dear ones by your grace. I seek this boon of you.
The story in brief is -
When Indra the lord of celestials asks Rama for a boon, Rama requested Indra to bring back to life, all the monkeys who had lost their life in the battle. Indra grants the boon and all the dead monkeys are restored to life. The celestials disperse and the whole army of monkeys enjoy their well-earned rest.
There are plenty of resurrection stories mentioned in Hinduism. Some of them are below:
- Ganesha brought back to life after beheading.
Lord Ganesha's head was chopped off and it was replaced by Lord Vishnu with head of a bull elephant which was sleeping with its head sleeping towards north direction.
O sage, at the gaze of the Saturn, the head of the child was cut off and Saturn closed his eyes at once, Looking downwards and stood there. 
He (Vishnu )took out the trunk of the child from the lap of Parvati, kept him in his lap and joined the elephant head with the trunk of the child. 
The lord who has been the form of Brahman applying the divine knowledge uttered humkara brought back the child back to life. 
In this way, Lord Ganesha attained was brought back to life by Lord Vishnu.
- The bull elephant whose head is cut off.
In the above story, Lord Vishnu beheads a bull elephant's head using his Sudarshana Chakra. That elephant was also brought back to life after being prayed by cow elephants. This story is explained in this answer by @Arya.
Thereafter, finding all the people fainted Vişņu mounted on Garuda, went to the northern direction and reached the bank of Puspabhadra river. 
Reaching the bank of Puspabhadra river, he found an elephant enjoying the company of cow elephants. 
Lord Visnu delightfully cut-off his head by using Sudarsana-cakra which was soaked in blood and kept it over Garuda and was quite pleasant to look at. 
There after, the wives and the children of the bull elephant pray Lord Vishnu. Lord Vishnu pleased with the prayer of the elephants restores the life of the bull elephant and bestows a life for Kalpa.
The lord who is all knowledgeable, touched the body of the elephant with his lotus-like feet and said, "O elephant, you will remain alive with your family for a kalpa." [20-21]
- The resurrection of Vānaras from the Rāmāyaṇa.
As explained in the other answer to this question, Valmiki Rāmāyaṇa Yuddha Kanda sarga 120 mentions the resurrection of Vānaras. The dead Vānaras were brought back to life by Lord Indra after being asked by Lord Sri Rāma.
O the bestower of honour! I wish to see all those monkeys, who for my sake, were removed from their sons and wives, be delighted at heart. They were all valiant, proving their energies and did not take their death into account. They made their strenuous efforts and died. O Indra! Restore their lives to them. Those monkeys - who counted death as nothing and who were fond of doing favour to me - let them get re-united with their near and dear ones by your grace. I seek this boon of you. O bestower of honour! I wish to see the monkeys and bears, free from wounds and pains, with augmented strength and valour. [6-9]
Indra restores the lives of the Vānaras and bears who have lost their lives in the battle.
Let all those monkeys and bears along with long-tailed monkeys, who have been killed in battle, by the demons and whose heads and arms have been severed, be alive and active again. 
All of them, full of delight, will get re-united with their friends, relatives, kinsmen and family members. 
- Bharata's son Puṣkala brought back to life by Hanuman.
This story which is not very famous is also associated with the Rāmāyaṇa. But this is told in the Chapter 43-44, Patala Khanda of Padma Purana. This happens when Sri Rāma sets his sacrificial horse loose when he performed a horse sacrifice. Sri Rāma appoints his younger brother Shatrughna as the protector of the horse. A huge army is also sent along with the horse. Bharata's son Puṣkala is also one of the warriors who is sent along with the horse.
As a part of its journey, the horse enters the kingdom of King Viramani. He doesn't agree to surrender the kingdom to Sri Rāma. Finally Puṣkala, son of Bharata resolves to fight with him. He wins the battle. But the king Viramani is a devotee of Shiva. So, he himself has arrived at the battle field along with his forces. He sends Virabhadra to fight with Puṣkala. He is killed by Virabhadra after a fierce fight.
Virabhadra, with the Khatvanga in his hand cut off became extremely angry and shattered the warrior's chariot. Having broken the chariot of the warrior and having (thus) made him a foot-soldier, he fought with the magnanimous Puṣkala in a close fight. That very mighty Puṣkala abandoning the chariot shattered to pieces by him (i.e. Virabhadra), hit Virabhadra with his fist. [---------] Then Puṣkala who was extremely angry seizing Virabhadra by the neck struck (i.e. threw) him on the ground. The very mighty Virabhadra distressed by that stroke seized Puṣkala by the foot and repeatedly shaking him and throwing him on the ground the very mighty one cut off his head with blazing ear-rings with his trident. The very strong Virabhadra having killed Puṣkala roared. That roaring follower of Siva frightened great warriors. When Puṣkala fell in the battle, there was a very great wailing. All the very proficient men became frightened on the battlefield. They informed Satrughna that Puṣkala killed by Virabhadra, the attendant of Siva, had fallen on the battlefield. The great hero, the mighty (Satrughna) having thus heard about Puṣkala's being killed in the battle, was very much grieved and due to great grief trembled. [28b-42a]
Thereafter Shatrughna also fights with Lord Shiva but falls unconscious in the battle. Enraged with the fall of the devotees of Sri Rāma, Hanuman starts a fight with Shiva. He destroys the chariot, flags, trident of Lord Shiva. He hits Shiva with trees and mountains. By this act, Shiva enrages and said
(He said): "O mean monkey, you are (now) killed. Flee and go (away) from the battle. In a moment I shall destroy your life with (this) pestle." Seeing the pestle discharged by the angry Siva the monkey remembering Visnu (i.e. Rāma) avoided it very speedily. That pestle made of strong iron and discharged by Siva, fell down. Having torn off the entire earth it went to the nether world. Hanumat, the servant of Rāma, was extremely angry at that time. Taking a mountain in his hand he hit it on the chest (of Siva). While the lord of SatI (i.e. Shiva) was thinking of cutting off the mountain he was struck by the lord of monkeys with a tree having many branches. When he (i.e. Siva) was intent on cutting it off, he was hit with rocks (by Hanumat). The persevering Siva made up his mind to break those rocks. Just then he showered him (i.e. Siva) with trees and mountains. Having encircled the protector of the beings (i.e. Siva) with his tail he struck him again and again with stones, mountains, trees and strokes of his tail. Nandin was very much frightened, and even the moon was reduced to pieces. The very angry great god got very much confused. [22-29a]
He said to the lord of monkeys who every moment made him very much disquieted: "O follower of the protector of Raghus, you are blessed. You have today done a great feat, so that I am very much pleased with you. O you, having a great speed, I am not easily obtainable by means of charity or sacrifice or a little penance. Therefore, ask for a boon from me. [29b-32]
Hanuman laughing and speaking in fearless words, spoke to Shiva who was speaking like this:
O great god, by Ragunatha's grace I have everything. Yet I ask for a boon from you who are pleased with the battle. This our (hero) named Puṣkala has fallen down dead. So also Shatrughna, Rāma's younger brother, became unconscious in the battle. Many other heroes wounded with arrows have also fallen. Some are unconscious, some have sunk; (please) protect them along with your own attendants. Do that by which big ghosts, vampires and fiends, so also foxes and others, will not take them away and eat them up; and (do that by which) their bodies will not be split. By all means preserve all of them, till, having conquered the attendants of Indra, I bring (either) the Drona mountain or the herbs thereon, and bring back, per force, to life these heroes kept under (your) control. Here I go to take that Drona mountain on which life-restoring herbs exist. [34-40]
Hanuman goes to Drona mountain which is guarded by the attendants of Lord Indra. He goes and uproots the Drona mountain. Devatas were frightened by this act of Hanuman. Then Indra bows to him asks him why he arrived there and allows him to take required herbs. The devatas bow to Hanuman and he takes leave from there. He brought the medicine back to the battlefield. He joined the head and body of the dead Puṣkala and said
If I recognise (only) lord Raghava (i.e.Rāma) by the deeds of my mind, body and speech, then let him quickly revive by means of the medicine.". Puṣkala came back to life.
Hanuman then went to Shatrughna, put the medicine on his chest and said
O best Satrughna, revive. O you of great might and valour, why (i.e. due to what) did you become unconscious on the battlefield? If I being diligent (i.e. diligently) keep celibacy till (the end of) my life, then let this hero Satrughna revive in a moment.
No sooner did he utter these words then Satrughna came back to consciousness in a moment.
In this way, Hanuman revived Puṣkala, the son of Bharata.
- Kṛṣṇa and Balarāma rescuing their teacher Sāndīpani's son.
Balarāma and Kṛṣṇa studied at the ashrama of Sāndīpani. They served him with great obedience. They learned all the topics in a very short period of time.
SB 10.45.35-36 — O King, those best of persons, Kṛṣṇa and Balarāma, being Themselves the original promulgators of all varieties of knowledge, could immediately assimilate each and every subject after hearing it explained just once. Thus with fixed concentration They learned the sixty-four arts and skills in as many days and nights. Thereafter, O King, They satisfied Their spiritual master by offering him guru-dakṣiṇā.
SB 10.45.37 O King, the learned brāhmaṇa Sāndīpani carefully considered the two Lords’ glorious and amazing qualities and Their superhuman intelligence. Then, after consulting with his wife, he chose as his remuneration the return of his young son, who had died in the ocean at Prabhāsa.
They go to Yamaloka and ask Yamarāja to send their teacher with them. Yama rāja worships them elaborately and agrees with them. He sends the boy with them. They bring their teacher's son back to life.
SB 10.45.45 — The Supreme Personality of Godhead said: Suffering the bondage of his past activity, My spiritual master’s son was brought here to you. O great King, obey My command and bring this boy to Me without delay.
They take him to their teacher's ashrama and fulfill their guru-dakṣiṇā. Sāndīpani gives them permission to return to their home.
In this way, Kṛṣṇa and Balarāma brought their teacher's son back to life.
- Bali Chakravarti brought back to life by Śukrācārya. Bali chakravati was brought back to life when he was killed in a war with Indra. Śukrācārya brought him back to life. Due to this, Bali became faithful to Śukrācārya. Many Rakshasas were revived by Śukrācārya using Mrita Sanjivani vidya but all of them were slain by Vishnu. Bali Chakravarti was not slain. He is in Sutala loka along with his friends and allies.
parājita-śrīr asubhiś ca hāpito
hīndreṇa rājan bhṛgubhiḥ sa jīvitaḥ
sarvātmanā tān abhajad bhṛgūn baliḥ
Śukadeva Gosvāmī said: O King, when Bali Mahārāja lost all his opulence and died in the fight, Śukrācārya, a descendant of Bhṛgu Muni, brought him back to life. Because of this, the great soul Bali Mahārāja became a disciple of Śukrācārya and began to serve him with great faith, offering everything he had. [SB 8.15.13]
Above are some of the incidents of resurrection mentioned in the Hinduism.
Apputhi Adigal was one of the 63 Nayanars. He was very devoted to Thirunavukkarasar, who was another Nayanar. When Thirunavukkarasar went to home of Apputhi Adigal to meet Apputhi Adigal, he saw that Apputhi Adigal named his every asset as Thirunavukkarasu. Apputhi Adigal lately recognized that the person who came to his home was none other than Appar aka Thirunavukkarasar.
Apputhi Adigal requested him to have food at his home. When Thirunavukkarasar consented, Apputhi Adigal and his wife ordered his eldest son called Senior Thirunavukkarasu [Out of his devotion towards Thirunavukkarasar, Apputhi Adigal named his elder son as Mootha Thirunavukkarasu(Senior Thirunavukkarasu) and his youngest son as Ilaya Thirunavukkarasu(JuniorThirunavukkarasu)] to bring banana leaves from the garden. A snake bit him in the garden while trying to pluck banana leaves. So, he died there.
Behold my boon! I have been plied in this Holy task by my goodly and righteous parents!”
Thus he thought and ran to the garden. As he cut a broad and tender plantain-leaf
A dazzling snake bit him on his palm Causing him to fall down in pain and giddiness.
Apputhi Adigal hid the corpse of the boy and tried to serve food to Thirunavukkarasar. But Thirunavukkarasar wanted to apply holy ash to the eldest son, who just died. So, he requested Apputhi Adigal to order his eldest child to come back from the garden. Apputhi said he would not be available without mentioning the son's death. Finally, Apputhi revealed the truth to Thirunavukkarasar. Appalled, Thirunavukkarasar prayed to Shiva with the hymn Ondru Kolam Avar thevaram. The child was resurrected.
When Navukkarasar heard the death event of his child, he exclaimed: “
Great indeed is that you have wrought! Whoever had done like unto you?”
Then he rose up and walked to the moribund. He hymned a musical song which in its wake
Caused the flow of the grace of Lord; Thus he chased the venom away from the child's body.
Parikshit was revived by Lord Krishna. This is discussed in this post.
Lord Paraśurāma’s mother and brothers were revived by Jamadagni. This is described in Śrīmad-Bhāgavatam (Bhāgavata Purāṇa) 9.16.8.
Sage Bhrigu revived his wife. This is discussed in this post.
Daksha was revived by Lord Shiva. This is discussed in this post.