Yes Lord Parshurama has an important role in Vyasa's Mahabharata.
In Adi Parva the Rishis present in NaimishAranya asks Sauti about the region of Samanta Panchaka(where Kurukshetra war took place)
The Rishis said, 'O son of Suta, we wish to hear a full and circumstantial account of the place mentioned by you as Samanta-panchaya'.
To this Sauti replies how the region of Samanta Panchaya is related to Parshurama.
Sauti said, 'Listen, O ye Brahmanas, to the sacred descriptions I utter O ye best of men, ye deserve to hear of the place known as Samanta-panchaka. In the interval between the Treta and Dwapara Yugas, Rama (the son of Jamadagni) great among all who have borne arms, urged by impatience of wrongs, repeatedly smote the noble race of Kshatriyas. And when that fiery meteor, by his own valour, annihilated the entire tribe of the Kshatriyas, he formed at Samanta-panchaka five lakes of blood. We are told that his reason being overpowered by anger he offered oblations of blood to the manes of his ancestors, standing in the midst of the sanguine waters of those lakes. It was then that his forefathers of whom Richika was the first having arrived there addressed him thus, 'O Rama, O blessed Rama, O offspring of Bhrigu, we have been gratified with the reverence
thou hast shown for thy ancestors and with thy valour, O mighty one! Blessings be upon thee. O thou illustrious one, ask the boon that thou mayst desire.'
To this Parshurama replies
"Rama said, 'If, O fathers, ye are favourably disposed towards me, the boon I ask is that I may be absolved from the sins born of my having annihilated the Kshatriyas in anger, and that the lakes I have formed may become famous in the world as holy shrines.' The Pitris then said, 'So shall it be. But be thou pacified.' And Rama was pacified accordingly. The region that lieth near unto those lakes of gory water, from that time hath been celebrated as Samanta-panchaka the holy. The wise have declared that every country should be distinguished by a name significant of some circumstance which may have rendered it famous. In the interval between the Dwapara and the Kali Yugas there happened at Samanta-panchaka the encounter between the armies of the Kauravas and the Pandavas. In that holy region, without ruggedness of any kind, were assembled eighteen Akshauhinis of soldiers eager for battle. And, O Brahmanas, having come thereto, they were all slain on the spot. Thus the name of that region, O Brahmanas, hath been explained, and the country described to you as a sacred and delightful one. I have mentioned the whole of what relateth to it as the region is celebrated throughout the three worlds.'
Later in Adi Parva the Bhrigu dynasty of Parshurama is further explored.
Also later in the epic Parshurama gave sudarshan chakra to Krishna(during His time at His guru Sandipani).
This along with being teacher of Bhishma,Drona and Karna explains the role of Lord Parshurama in Mahabharata.
I will updating this part of answer if I find other incidents about Parshurama.
Now about your other questions.
But how Vishnu took the role of both Parashuram and Krishna in the same period. Where was Parashuram during the time of Mahabharatha?
This is not something new as two Avatars of Vishnu CAN exist at the same time. Lord Parshurama has an important role in Ramayana. Thus even in Treta Yuga both Avatars of Vishnu existed at the same time. As we know from Mahabharata,and as quoted above the last time Parshurama destroyed Kshtriyas was between Treta Yuga and Dwapar Yuga.
To your other question,
Bheeshmas age during Mahabaharatha was 120years and what was Parashurams age at that time?
Again, Parshuram is no normal human,He is Vishnu incarnate. Also He is one of the Chiranjavis (immortals). So comparing Bhishma( who was divine birth yet mortal) with An Avatar of Sri Vishnu is not fair.
What really happened to Parashuram?
As said above, the last time Parshurama destroyed Kshtriyas was between Treta and Dwapara Yuga. He seem to have retired in Treta Yuga resting in MahendraGiri mountain. That is until Dwapara Yuga arrived, as He continued to train Brahmins in Shastra Vidya. This included Drona and Karna(who hid his real identity). So right now he is still at Mahendra Giri mountain waiting for Kalki Avatar. As Lord Parshurama will train Kalki and will be His Guru.
Your last question.
Is he Chiranjeevi or not?
Yes He is a Chiranjivi. He is considered one of the seven Chiranjivis.
अश्वत्थामाबलिर्व्यासोहनुमांश्च विभीषण:कृपश्चपरशुरामश्च सप्तैतेचिरंजीविन:।
Ashwathaama Balirvyaaso HanumanshchaVibhishanaha Krupascha ParshuramaschaSaptaitey Chiranjivinaha
Some other lists include Rishi Markandya thus concluding the number of Chiranjivis to eight.
अश्वत्थामा बलिव्र्यासो हनूमांश्च विभीषण:। कृप: परशुरामश्च सप्तएतै चिरजीविन:॥ सप्तैतान् संस्मरेन्नित्यं मार्कण्डेयमथाष्टमम्। जीवेद्वर्षशतं सोपि सर्वव्याधिविवर्जित॥
Ashwathaama Balirvyaaso Hanumanshcha VibheeshanahaKrupaha Parshuramascha Saptaitey ChiranjivinahaSaptaitaan Samsmareynnityam MarkandeymathaashtamamJivedvarshshatam Sopi Sarvavyadhivivarjit