AFAIR, Though Parasurama was blessed with the axe from Vishnu, he is the indirect incarnation of Vishnu (meaning , with the spirit/power fo vishnu), but after being defeated/humbled by Sri Rama, he went to kailash. So, he himself must be Shiva/Rudra. Why are you de-nigrating one Rudra personified, being a saivite
?
Parasurama's words when defeated by Rama in the challenge of stringing Vishnu's bow,
"It will be apt of you to unloose that unsurpassed arrow, oh, Rama, as you have ethical commitment to wipe out demons as I wiped out
menacing kings, and should you unloosen that arrow now I wish to depart to the ethereal mountain Mt. Mahendra, a point of no return for me..." So said Parashu Rama to Dasharatha Rama.
"Oh, Kakutstha Rama, you are the lord of the triad of worlds, such as you are, you faced me down, and it is malapropos to say that this is a disgrace to me..."
Valmiki Ramayana - Bala Kanda
I guess, the leftout/products of Parasurama's adventure were the Jarasandha, Kamsa etc. Seems like, he didn't threaten the Dasaratha's lineage of kshatriyas too. Also, FYI, Parasurama was a mix of Brahmin and Kshatriya, and it is likely to be valourous against adharma.