Do I need to kill Paarthunax?

You do not need to kill Paarthurnax. But if you want to remain in good allegiance with the Blades, you will have to slay him. The perks include Esbern's blessing, recruiting new Blades members, and visiting Sky Haven Temple. If you want to remain in allegiance with the Greybeards, do not kill Paarthurnax. The perks include strengthening your shouts, finding word wall locations, and visiting High Hrothgar. You will also get a bonus cut scene after defeating Alduin that is worth watching. It really does come down to morals, in the end. The choice is yours.