Parasect's Quick Moves

He has Fury Cutter and Bug Bite both are same Bug type and Bug Bite is significally better with more dps and more eps

Indeed. Fury Cutter has no real advantage over Bug Bite.

Fury Cutter is the weekest and fastest move in game. The rounding mechanism prefers weeker moves, as it always rounds up. [floor(...)+1]
With 3 damage (and the rough estimation of equal Attackers att-Value and defenders def-Value) rounding up 3 of damage gives values of 3 to 4, average of 3.5, 16% average bonus due to rounding issues. This bonus is smaller with higher damages.
Attention: This estimation is really rough, it may differ greatly dependend on actual battle.

This makes Fury Cutter a little better, but still worse than Bug Bite.

Maybe in some setups, when dodging every quick move, the faster Fury Cutter can squeeze 5 attacks between two dodges, where Bug Bite only manages 4 attacks. In this case, Fury Cutter would be better.
It would need a sheet to calculate, if this is relevant.

Just some thoughts.

tldr
IMO FC is worse than BB. Seldom it could be better.

I would not power him up because Parasect has become to me an amazing gym prestiger, with the number of Vaporeons and Exeggutors in my area i have 3 Parasect with around 1k-1,2k CP (1 with bug bite and X-Scissor 1 with fury cutter solar beam and another with bug bite solar beam, all 85%+ IV), all of them are very useful, I personally don't see much difference between the 2 attacks but it's true that bug bite is slightly better on terms of dps and eps.

Bug Bite is better, but they are so close I use both. Paras are so common where I live, I have an army of over a dozen Parasect that I use for training. After one fight, I can switch to another and wait to heal later when I am collecting pokestops and getting a l ot of purple potions that I would otherwise throw away for space.

My go-to Exe basher..especially friendly gyms needing lower cp-attackers.
I like XScissor the most, best move in the game because of the short cool down often you can dodge a confusion and use it with time to dodge the next.
Bite supposedly has higher energy created, (along with more damage of course) that gives it the higher rating..but cutter is just fine.

Back when the game came out, Fury Cutter had twice the eps it has now... it charged the special move quicker than any other fast move in the game. After they nerfed it, I got rid of my Parasects with FC and kept the BB's.