Parashurama has Vedic roots. His name was Rama Jamadagnya and he was the composer of a Sukta of Rigveda Samhita. According to Rgveda Sarvānukramaṇi, Rāma Jāmadagnya (Rāma the son of Jamadagni) was the composer of Rgveda Sukta ( Aprisukta) 10.110 of 11 verses in Triṣṭubh Chandas. It is as Rāma Jāmadagnya rather than “Paraśurāma” that he is primarily known in Vālmīki Rāmāyaṇa & Mahābhārata.
Paraśurāma was a great Rishi. If he beheaded his mother, it was only because he was bound by the words of his father. The first boon he asked his father was the restoration of his mother's and brothers' lives and complete forgetfulness of the incident. That was his love for his mother. If Paraśurāma had opposed the words of his father, Jamadagni's anger would not have been appeased and he would have cursed Paraśurāma along with the other brothers. What Paraśurāma did was very tactical. He appeased Jamadagni and restored the lives of his brothers and mother.
Paraśurāma rebelled against unjust rulers of his time like Kartavirya Arjuna who unjustly slaughtered unarmed Jamadgni and stole their cow. That does not mean he hated the Kshatriyas. Many prominent Kshatriyas like Bhishma were trained in arms and warfare under him.
When a grave injustice was done to the Kshatriya princess Amba and nobody could oppose the might of Kurus who came to the rescue of Amba and ensured her justice? Yes, it was none other than Paraśurāma! He helped everybody who came to him.
Paraśurāma was a great warrior. All the greats of Mahābhārata like Bhishma, Drona, and Karna were trained under him. He conquered every region. Yet, he desired nothing. He donated everything to Kashyapa and retired to live the life of an ascetic in Mahendragiri hills.