The story of Lord Parashurama receiving axe from Lord Shiva is described in Brahmanda Purana. It is described in part 2 chapter 24: "Penance of Parashurama; His acquisition of the knowledge of astras". You can download this book here. Here is how Brahmanda Purana narrates the event:
Having traversed thus all over the earth at the behest of god Sambhu, the descendant of Bhrgu went again to that land where he had been living before penance.
After going there, O king, and staying there itself, he devoutly worshiped lord Umapati by means of austerities and holy Observances.
It was in the meantime, O king, that a long-drawn war took place between the Devas and the Asuras causing horripilation.
Thereupon the excessively powerful Asuras defeated the gods and acquired the glory and riches of the Devas. They had nothing to fear from anywhere.
The Devas who had been defeated in the war— Vasava (Indra) and all the others whose assets and magnificent glory had been seized by their enemies sought refuge in Sankara.
After propitiating the lord of the universe by means of obeiscence and eulogies about his victory, the Devas requested the Pinaka wielding Lord (Siva) to destroy the Asuras.
Thereupon, O king, Sambhu, the bestower of boons on the Devas, promised to them the slaughter of the Danavas and spoke thus to Mahodara.
In the southern sector of Himalayas, a sage‘s son of great penance, named Rama, is performing penance directed towards me. He is very brilliant.
Go there now itself, O Mahodara and convey my behest to him. Even as he is performing penance let him be brought here without delay".
On being ordered thus, Mahodara bowed down to the lord after saying "So be it”, and went with the velocity of the Wind to the place where Rama was settled down.
After reaching that spot and seeing Rama the greet sage performing penance, he humbly spoke these words.
“O excellent one among the descendants of Bhrigu Lord Sambhu wishes to see you. I have come here at his behest. Hence, come to the presence of his lotus-like feet".
On hearing his words, BhArgava got up quickly, received his behest approvingly with his head (bowed down). and replied—“So be it".
Thereupon, Mahodara with great speed led Rama immediately to a place by the side of Sambhu on the Kailash, the excellent mountain.
The most excellent scion of the family ‘Bhrgu saw Sankara who endears himself to his devotees. accompanied by the goblins as well as Indra and other immortal beings (ie. gods).
The lord was being eulogised by Narada and another ascetic: whose wealth was penance. He was being attended upon by the Gandharvas who were singing, and by the groups of Apsaras who were dancing. The lord of the Devas had the hide of an elephant for his garment and had smeared the entire body with the holy ashes. The three~eyed moon-crested Lord had the tawny-matted hair over his head.
He had adorned himself with serpents serving the purpose of ornaments. His lips and arms were long. He was gentle. His lotus-like face appeared delighted. O king, he was seated on a golden slap in the midst of the assembly of the Devas.
The most excellent one among the descendants of Bhrgu approached the lord of the Devas with palms joined in reverence. His entire body had hairs standing on end as a result of his vision of Srikantha. He went near Hara with his body drenched in tears of joy. He was excited due to delight, he uttered the words, ”Obeisance to you, O lord of the Devas" several times. Finally, he fell down at the feet of the enemy of the three Puras (ie. god Siva).
Amidst the groups of the Devas who were looking on, ‘the delighted lord Siva lifted up the most excellent scion of the family of Bhrigu. With an enthusiastic smile and eagerness, he spoke these sweet words to Rama whose lotus-like face appeared pleasant.
Then Girvanas (Devas) have been attacked by the hosts of Daityas. They were dislodged from their abodes. Unable to kill them they have approached me.
Hence, at my behest I! well as with I desire to do something pleasing to the Devas, O Rama, kill the entire hosts of Daityas. You are considered capable of it by me”.
Then Rama bowed down to Sarva. Even as all the Deva: stood listening, Rama spoke these words very humbly with palm joined in reverence.
“O lord, you are omniscient. You are the immanent soul of everyone. Hence, is not (what I am going to say) known to you (already) Still, listen to the words I submit.
If it is impossible for Sankara and other Devas to kill the asuras how can they be killed by me single handedly ?
I am ignorant of miraculous missiles, O lord of the Devas. I am not an expert in warfare. How can I kill all the enemies of the Deva: without any weapon whatsoever ?“
On being told by him thus, the lord of the Devas transferred to that noble-souled Rama white lustrous splendor of Shiva, full of miraculous missiles, as brilliant as the fire of Death.
After handing over his own axe that could overpower all other weapons, the Lord delighted within himself, said to Rama even as the Devas stood listening.
“Due to my grace, O gentle one, may there be sufficient power in you enabling you to kill all the enemies of the Devas, and be unassailable In all your enemity.
Go ahead with this weapon alone and tight with your enemies. You yourself will duly come to know and possess the skill in war."
Vasistha said :
On being instructed thus by Sambhu, Rama bowed down to him, and accepted the axe pertaining to Siva, and was bent upon the slaughter of the enemies of the Devas.
Thereupon, Rama who was born of a part of splendor of Vishnu and who was equipped with devotion to Rudra shone like the splendor of the sun accompanied by his brilliance.
Permitted by the three-eyed lord and accompanied by all the Devas he went ahead to slay the Asuras. He. moved to fight with them till the end.
Thereupon, once Again there broke out the war of the Devas with the Asuras, O king. They were endeavoring to conquer the three worlds. Hence the war was extremely terrible.
Then Rama of mighty arms became infuriated in that excessively terrifying battle. He killed the great Asuras with his axe.
Killing the Daityas in thousands by means of strokes resembling the thunderbolt, the infuriated Ram moved about in the battlefield like another god of 'Death.
After killing all the Daityas in a trice, Rama the most excellent one among those who strike (with a weapon) exterminated them and gratified all the Devas.
Everyone among the Daityas and the Danavas was being struck at by Rama. Those who survived the slaughter saw Rama everywhere and became frightened.
When the entire host: of the Asuras were either killed or routed, the Devas took leave of Rama and went again to the Heaven.
After killing the son of Diti and bidding farewell to the Devas, Rama resorted to his hermitage with his mind attached to the activities of penance.
That rage of great intellect made the image of a hunter of deer representing Sambhu and devotedly worshiped it in his hermitage, with perfect self-control. He duly appropriated him with fragrant incense, beautiful flowers, food offerings, obesicences, and eulogies.