Story of Parashurama is mentioned in many Puranas and in the Mahabharata. I'm quoting a section of the Chapter 116 and 117 of the Vana Parva of Mahabharata.
Parashurama was the son of Rishi Jamadagni and Renuka.
There was once a King named Kartavirya Arjuna of the Haihaya dyanasty. Rishi Jamadagni and Renuka offered hospitality to Kartavirya Arjuna and his army at their ashram. Jamadagni had a Kamadhenu (wish fulfiling) cow that produced food etc for the King and his army. Kartavirya on seeing the cow's utility stole her away from the sage and his wife.
Once, however, O lord, when his sons had gone out as before, the
valourous son of Kartavirya, the lord of the country near the shore
of the sea, came up to the hermitage. And when he arrived at that
hermitage, the wife of the saint received him hospitably. He,
however, intoxicated with a warrior's pride, was not at all pleased
with the reception accorded to him, and by force and in defiance of
all resistance, seized and carried off from that hermitage the chief
of the cows whose milk supplied the sacred butter, not heeding the
loud lowing of the cow. And he wantonly pulled down the large trees
of the wood.
When Parashurama came to know he went to Haiheya and attacked the King and decimated his army. After killing Kartavirya, by cutting off his 1000 arms Parashurama brought the cow back to the ashram.
When Rama came home, his father himself told him all that had
happened. Then when Rama saw how the cow was lowing for its calf,
resentment arose in his heart. And he rushed towards Kartavirya's
son, whose last moments had drawn nigh. Then the descendant of Bhrigu,
the exterminator of hostile heroes, put forth his valour on the field
of battle, and with sharpened arrows with flattened tips, which were
shot from a beautiful bow, cut down Arjuna's arms, which numbered a
thousand, and were massive like (wooden) bolts for barring the door.
He, already touched by the hand of death, was overpowered by Rama, his
Kartavirya's sons were enraged at the murder of their father so they attacked a defenseless Jamadagni at his ashram when Parashurama and his other sons were away. They killed the rishi and stole the cow and its calf.
Then the kinsmen of Arjuna, their wrath excited against Rama, rushed at Jamadagni in his hermitage, while Rama was away. And they slew
him there; for although his strength was great, yet being at the time
engaged in penances, he would not fight. And while thus attacked by
his foes, he repeatedly shouted the name of Rama in a helpless and
piteous way. And, O Yudhishthira, the sons of Kartavirya shot
Jamadagni, with their arrows, and having thus chastised their foe,
went their way. And when they had gone away, and when Jamadagni had
breathed his last, Rama, the delight of Bhrigu's race, returned to the
hermitage, bearing in his arms, fuel for religious rites. And the
hero beheld his father who had been put to death.
Parashurama on seeing his father dead made a vow to kill all Kshatriya generations as they sprung up.
And Rama, the conqueror of hostile cities, cremated his father on the funeral pyre, and vowed, O scion of Bharata's race, the slaughter
of the entire military caste, and of exceeding strength in the field
of battle, and possessed of valour suited to a heroic soul, and
comparable to the god of death himself, he took up his weapon in
wrathful mood, and singlehanded put Kartavirya's sons to death. And, O
chieftain of the military caste, Rama, the leader of all capable of
beating their foes, thrice smote down all the Kashatriya followers of
Kartavirya's sons. And seven times did that powerful lord exterminate
the military tribes of the earth. In the tract of land, called
Samantapanchaka five lakes of blood were made by him.
Canto 9 Chapter 15 of Srimad Bhagvatam gives reasons why Parashurama killed the kshatriyas.
SB 9.15.14 — Learned scholars accept this Paraśurāma as the celebrated
incarnation of Vāsudeva who annihilated the dynasty of Kārtavīrya.
Paraśurāma killed all the kṣatriyas on earth twenty-one times.
SB 9.15.15 — When the royal dynasty, being excessively proud because
of the material modes of passion and ignorance, became irreligious and
ceased to care for the laws enacted by the brāhmaṇas, Paraśurāma
killed them. Although their offense was not very severe, he killed
them to lessen the burden of the world.
Chapters 74-76 of Bala Kanda of Ramayana describes a meeting between Rama and Parashurama. After Rama breaks the bow of Shiva at Sita Swayamvar,
Parashurama accosts the wedding party and challenges Rama to string the bow of Vishnu. Rama does so and asks Parashurama what should be his target as an arrow stretched on the bow should not be wasted. Parashurama allows him to destory his tapasya merit and returns to Mt Mahendra.
In that catastrophic darkness, that sand-muffled military of king
Dasharatha has seen the son of Sage Jamadagni, namely Bhaargava Rama,
the subjugator of kings of kings.
"Oh, valiant Rama of Dasharatha, your valour is bruited as a
sensational valour, and sensational is your smashing of Shiva's bow,
That way, smashing of that particular bow of Shiva is wondrous and
even unimaginable... on hearing that alone I happened upon you,
bringing another outranking bow...
This alone is that catastrophic bow received through Sage Jamadagni...
flex it with an arrow on bowstring stretching up to your ear, and in
that way show your capability... Thereby, on examining your strength
in your taking aim with this bow, and should you be deservedly
valorous, I will give you a combative duel..." So said Parashu Rama to
Raghava, the nimble-handed vanquisher of his opponents, speaking thus
in high dudgeon expropriated that estimable weapon, namely the longbow
of Vishnu, from the hand of Bhaargava Rama, along with the long-arrow
that is already fitted on it...
Then, when Rama is ready to take aim with the arrow on that
inscrutable longbow, and when the worlds are being rendered as
insentient, then that Rama of Jamadagni is rendered vigourless and he
stared up at Rama of Dasharatha.
Rama of Jamadagni is calloused as his vitality is subdued by the
radiance of that lotus-petal eyed Rama of Dasharatha, and he spoke to
Rama of Dasharatha, slowly and softly.
"Once, when I donated entire earth to Sage Kashyapa, Kashyapa told me
'uninhabitable is my domain, viz., this earth for you...' thus...
"Such as I was, in my observance of my mentor Kashyapa's order I do
not spend nights on this earth from then afterwards, oh, Kakutstha
Rama, as I made over this earth for Kashyapa, indeed... Thereby oh,
valiant one, it will be inapt of you to impair this motility of mine,
oh, Raghava, I will depart with the speed of cerebration to Mt.
Mahendra, a par excellent mountain... But I triumphed over matchless
realms of heavens with my ascesis, oh, Rama, you may hash them up with
that irreversible arrow... let there be no time-lag... I have realized
your touch of nature as that of the Immutable Supreme Being, God of
Gods, the Exterminator of the demon Madhu, namely Vishnu, by the touch
of your handling that bow... oh, enemy-inflamer, blessedness alone
While Rama of Jamadagni is speaking that way, that venturesome and
blessed Rama of Dasharatha shot off that nonpareil arrow from that
longbow of Vishnu. On seeing all of his realms of heavens are
shot-blasted by Rama of Dasharatha, Rama of Jamadagni vanished in a
trice to Mt. Mahendra, the heavenly mountain.
Parashurama is referenced numerous times in Mahabharata. He was the guru of Bhishma, Drona and Karna. He also fought with Bhishma which is described in this answer.
Adi Parva Ch 101
And Ganga said, 'O tiger among men, that eighth son whom thou hadst
some time before begat upon me is this.
And all the weapons known to the powerful and invincible Rama, the son of Jamadagni are known to this thy illustrious son of mighty
arms. O king of superior courage, take this thy own heroic child
given unto thee by me.
Adi Parva Chapter 132:
O king, it was about this time that Drona heard that the illustrious Brahmana Jamadagnya, that slayer of foes, that foremost one among all
wielders of weapons, versed in all kinds of knowledge, had expressed a
desire of giving away all his wealth to Brahmanas. Having heard of
Rama's knowledge of arms and of his celestial weapons also, Drona set
his heart upon them as also upon the knowledge of morality that Rama
'O thou of ascetic wealth, returned Rama, 'My gold and whatever other
wealth I had, have all been given away unto Brahmanas! I have now my
body only and my various valuable weapons left. I am prepared to give
either my body or my weapons. Say, which thou wouldst have! I would
give it thee! Say quickly!'
"Drona answered, O son of Bhrigu, it behoveth thee to give me all thy weapons together with the mysteries of hurling and recalling
"Saying, 'So be it,' the son of Bhrigu gave all his weapons unto Drona,--indeed, the whole science of arms with its rules and
Santi Parva Chapter 2
When Drona had answered thus, Karna, having worshipped him, obtained
his leave, and proceeded without delay to Rama then residing on the
Mahendra mountains. Approaching Rama, he bent his head unto him and
said, 'I am a Brahmana of Bhrigu's race.' This procured honour for
him. With this knowledge about his birth and family, Rama received him
kindly and said, 'Thou art welcome!' at which Karna became highly
Parashurama was present in the Kaurava court when Krishna came to arrange a peace treaty with Duryodhana. He tried to counsel Duryodhana and narrated the story of Nara Narayana as referenced in Udyoga Parva Ch 128
"Vaisampayana said, 'Hearing these words uttered by the high-souled
Kesava, all the persons who sat in that assembly remained silent,
their hair standing on their ends. And all the kings thought within
themselves that there was no man who could dare reply to that speech.
And seeing that all the kings sat silent, Jamadagni's son (addressing Duryodhana) then said these words in that assembly of
Kurus, 'Listen confidingly to my words illustrated by an example,
and seek thy own good if my speech recommends itself to thee.
Chapter 39 of Harivamsa Parva of the Harivamsa describes a meeting between Krishna Balarama and Parashurama when they are fleeing from Jarasandha. I am only quoting the paragraph when they see him. The entire story is given in the link.
(vaishampAyana said) Saying this, both sa~NkarShaNa (balarAma) and
kR^iShNa, competent, proceeded on the southern route, without any
anxiety. Those two, capable of taking forms they desire, traveled in
many countries in the southern direction. The south bound valiant
balarAma and kR^iShNa happily went along the beautiful places behind
the sahya mountains.
Under that tree, ( both together saw) the sage, sparkling with the
wealth of penance, carrying the axe on his shoulder, wearing twisted
locks of hair and bark garments. (Both together saw) the sage,
sparkling like yellow flames of fire, with a splendour similar to sun,
who caused the death of kShatriya-s, imperturbable, appearing like
ocean, personified. (Both together saw the sage) seated in a posture,
having worshipped the sacred fire and taken ritual bath thrice as
ordained, like the preceptor of deva-s. (Both together saw the sage)
residing on the mahendra mountain, preparing to milk the white
kAmadhenu, the divine cow with its calf. Both together saw rAma,
bhArgava, the tireless, eternal sage without any decline seated like
the sun on mount mandara.
Seeing the sage, both the valiant sons of vasudeva, sparkling like the
sacrificial fire, worshipped his feet as ordained.
Parashurama was seen by the Pandavas and some sages on Mt Mahendra when the latter were on their Tirtha Yatra. However I can't find any references of him having fought the Pandavas sons.
"Vaisampayana said, 'Then on the fourteenth day of the moon, the mighty-souled Rama at the proper hour showed himself to those members
of the priestly caste and also to the virtuous king (Yudhishthira) and
his younger brothers. And, O king of kings, the lord together with
his brothers, worshipped Rama, and, O most righteous of the rulers of
men, the very highest honours were paid by him to all those members of
the twice-born class. And after worshipping Jamadagni's son and having
received words of praise from him, at his direction he spent the night
on the Mahendra hill, and then started on his journey towards the