Akritavrana is the son of the sage Shanta. He was rescued from a tiger by Parashurama, back before the major events of Parashurama's life like Kartavirya Arjuna killing Jamadagni and killing the kings of the world and all that. He has followed Parashurama ever since. Here is how this excerpt from the Brahmanda Purana describes it:
As he proceeded ahead, Rama who was induced by his previous actions controlling his future, entered a cave in the Himalaya mountains in a dense forest. There he saw a boy, the son of a Brahmana, crying like one who was extremely frightened as he was chased by a tiger. He was (as if) holding his vital airs (with great difficulty).
On seeing him, Rama had his heart overwhelmed with sympathy and he became excited due to his eagerness to save him. Saying “Stop, stop' loudly, he pursued that tiger. Chasing him with great speed, the scion of the family of Bhirgu, caught up the extremely terrible tiger in the forest after a long time. That Brahmana boy who was pursued by the tiger, fled with great fright for his life. He fell down near a cave in the forest. Rama had his eyes turned red due to anger. With a desire to redeem the Brahmana boy he took up a small blade of grass and chanted mantras over it as though it were a missile of Kusa grass. In the meantime the hyaena (tiger) of great power ran after the Brahmana boy. On seeing him fallen, he roared loudly making the heaven and earth tremble. With the fire of his missile, he burned the tiger who was striking with the tips of his claws. He thus released the Brahmana boy who was not at all wounded. That sinful (hyena) whose body had been burned completely due to the Brahminical fire, assumed the body of a Gandharva. Rising on to the sky, he spoke thus to Rama with great respect. O sage, formerly, as I had been cursed by a Brahmana I attained the state of a tiger. Redeemed now from the effects of that curse, I am going to heaven'.
When he went away hastily after saying thus, Rama who was wonderstruck sympathetically lifted up the fallen Brahmana boy. Sitting near the Brahmana boy he said the following words - 'Do not be afraid', O king. Slowly he stroked his limbs, reviving his life (and spirits). Lifted up thus by Rama, he opened his eyes and looked around. He saw the most excellent one among the members of the family of Bhirgu in front of him. He became surprised on seeing the tiger reduced to ashes. With his fear dispelled he said thus--'Who are you, please? How did you come here? By whom was this tiger reduced to ashes the hyena of terrifying shape and size, who almost appeared like another god of death and attempted to kill me? My mind is highly deluded and excited due to fear, O sage of great intellect. Although it has been killed, all the quarters appear to be filled with the tiger even now. I consider you to be everyone-my father, mother, friend and preceptor since you brought me back to life when I had met with the greatest of mishaps. There was a certain excellent sage of great penance called Santa. I am his son. Desirous of visiting holy centres, I went to the village of Salagrama. From there I proceeded to the mountain Gandha madana, desirous of visiting the holy penance grove of Badarika resorted to by many groups of sages. Desirous of going further, I lost my way on the Himàlaya mountain. Entering a fascinating forest, I got confused while viewing the different regions. I went a distance of a Krosa (three Kms) towards the east. As ill luck would have it, I was afflicted with fear and I had to run. When I had fallen down I have been lifted up now from the ground with great lovc and mercifulness by you like a son, by his father. Thus my story has been wholly narrated to you'....
On being asked thus by him, O leading king, Rama narrated his story entirely to him in due order. Mutually conversing thus they were delighted very much. He did not stay there for a long time thereafter. Then he desired to proceed ahead. Followed by him, Rama came out of that cave with great joy and started towards the residence of his parents. Since he was not wounded although he was caused to fall on the ground by the tiger and since he was saved by Rama who killed the tiger, the name of that Brahmana boy became well known on the Earth as Akritavrana (not wounded), O leading king. Ever since then, he became a great friend and follower of Rama in every situation like a shadow on the ground when there is sunlight.