Lord Parashurama is the sixth incarnation of Vishnu. Lord Parashurama was born in Treta Yuga on the day of Vaishakh Shukla Tritiya.
He was born to rishi Jamdagni (जमदग्नि) and Renuka (रेणुका). In Mahabharata, Renuka is mentioned as daughter of a King Prasenjeet (प्रसेनजित) and in Bhagavata Purana, she is mentioned as daughter of Rishi Renu
स प्रसेनजितं राजन्नधिगम्य नराधिपम्।
रेणुकां वरयामास स च तस्मै ददौ नृपः ।। – Mahabharata Vana parva 116.2
रेणो: सुतां रेणुकां वै जमदग्निरुवाह याम् – Bhagavata Purana 9.15.12
Renuka gave birth to five sons: Rumanvan (रुमण्वान्) , Sushen (सुषेण) , Vasu (वसु) , Viswa Vasu (विश्वावसु) and the youngest being Parashurama (परशुराम)
ततो ज्येष्ठो जामदग्न्यो रुमण्वान्नाम नामतः।
आजगाम सुषेणश्च वसुर्विश्वावसुस्तथा ।। – Mahabharata Vana Parva 116.10
Why did Parashurama kill his mother?
One day when Renuka went to the river, she saw King Chitraratha (चित्ररथ) of kingdom Martikavat (मार्तिकावत) . She was captivated by the sight for a moment as a result of which she lost her concentration and devotion to her husband.
When she entered the Ashram Jamdagni realized the events which has happened. Jamadagni got furious with Renuka and ordered his elder four sons to kill Renuka Devi but all of them refused to kill their mother.
Jamadagni, cursed his four sons because the disobeyed him and they lost their sense of mind due the curse. Then Jamadagni called his fifth son Parashurama and ordered him to behead Renuka Devi. Parashurama immediately obeyed his father and beheaded his mother with his axe.
Jamadagni was pleased by Parashurama’s devotion and obedience towards him. He then offered a boon to Parushurama, who wisely asked for his mother and brothers to be brought back to life. Jamadagni was impressed by Parshuram’s intelligence and brought Renuka and her four sons back to life.