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Background

You'll receive this quest if you talk to Esbern or Delphine (both usually located at Sky Haven Temple) during or after the main quest The Fallen.


Walkthrough

The Blade you talk to will tell you that Paarthurnax used to be one of Alduin's chief lieutenants, and that he was responsible for "terrible atrocities" during the Dragon War. The Blade will then inform you that "justice demands that [Paarthurnax] die for his crimes," and you'll be tasked with killing him.

You'll then have to make a choice. If you kill Paarthurnax, then the Greybeards will no longer deal with you, and you won't be able to call a truce for the civil war during the main quest The Fallen. If you don't kill Paarthurnax, then the Blades will no longer deal with you. If you don't trigger the quest at all, then the game will treat your actions as if you had triggered the quest and decided not to kill Paarthurnax.

Paarthurnax is a frost dragon, but he's not especially more difficult than a regular frost dragon, and so you shouldn't have any trouble killing him. Paarthurnax won't have anything special on his corpse, just the regular Dragon Bones, Dragon Scales, and gold.