Should I Kill Parallax Skyrim?

Solution: Attack and kill him. This will cause the quest to reappear in the journal. However, in some cases Paarthurnax will remain invincible even after the quest has disappeared, preventing further progress in the Blades storyline.

What happens if you don't want to kill Paarthurnax?

You can ignore that quest and continue. Killing paathurnax will fix your status with the Blades, but the Greybeards will no longer help you. Not killing will cause the Blade to not help you.

Is there any downside to killing Paarthurnax?

Cons: You will be hunted by the community. You will not escape. You have to live with the fact that you killed the 1 dragon that helped you by betraying his brothers and waited thousands of years to aid the one being in the whole planet that can destroy him. The greybeards hate you.