Paarthurnax (Quest)

Background
Delphine and Esbern learn that Paarthurnax is the leader of the Greybeards. They inform the Dragonborn of Paarthurnax's former position as the lieutenant of Alduin during the Dragon War and claim he was responsible for many unspecified atrocities against humankind. Bound by their oaths as Blades, they refuse the Dragonborn further assistance until Paarthurnax is slain. This quest is optional; it does not need to be completed in order to finish the main quest.

Overview
This quest is optional, but in order to continue the Blades storyline, it must be completed before the conclusion of the Main Quest. However, doing so will prevent any further communication with the Greybeards, including the radiant quest to locate Words of Power. The conclusion of the main storyline slightly differs depending on whether or not Paarthurnax is still living. Working with all factions prior to engaging Paarthurnax runs the risk of making the quest impossible to complete.

Killing Paarthurnax
Defeating Paarthurnax is not especially difficult as he displays only the usual powers common to all dragons and at this point the player must have access to both the Dragonrend shout and the Dragonbane sword.

How to avoid killing Paarthurnax (PC only)
It is possible to complete this quest without needing to slay Paarthurnax by typing this command into the console:

This way, Paarthurnax will remain alive, the Greybeards will still be hospitable, and the Blades will talk once more. The speech options regarding Paarthurnax's termination are still available when speaking with Arngeir, however.

If this quest is received during "Season Unending" and not completed before the end of the Main Questline, it will disappear from your list of active quests.

The console command  can also be used to avoid killing Paarthurnax, which allows the Blades to be killed instead.

Trivia

 * When speaking to Arngeir regarding the Blades' request, he will tell the Dragonborn about Paarthurnax's reason for siding with Alduin and later, his rebellion which brought to Alduin's first demise. There is the option to tell Arngeir that Paarthurnax must die or that he is going to be spared. This will affect the conversation between Esbern and Arngeir in "Season Unending," during the negotiation of a treaty.
 * Should the Civil War questline be complete (thus bypassing "Season Unending"), Delphine and Esbern will remain in Sky Haven Temple. However, they remain in disapproval and the quest disappears from the log.
 * There is no actual way of denying the request of the Blades, other than simply abstaining from doing it.
 * Without Arngeir, it is still possible to get radiant quests for words of power by shouting in a town or some other inhabited place and waiting for a Letter from a Friend.
 * This quest is still possible to complete after completing "Season Unending" by depleting all of Paarthurnax's health then quickly fast traveling to Sky Haven Temple.
 * It is possible to kill Paarthurnax directly after the quest "Alduin's Bane," since he is no longer an essential character. Once he is dead, traveling to Sky Haven Temple and talking to Esbern will get him to introduce the Dragonborn to the quest to kill Paarthurnax. At that point, the Dragonborn can tell him Paarthurnax is already dead. Note that this will not interfere with the Peace Treaty mission at High Hrothgar later.
 * If one has already completed the Civil War questline and then receives the "Paarthurnax" quest from Delphine, completing Dragonslayer and then talking to Arngeir and saying that they won't kill Paarthurnax will make the quest disappear from the quest log; it is not categorized by the game as "failed" (and as such would appear in the finished quests list), but simply disappears. This means that if the player should change their mind, they need to reload a save from before talking to Arngeir.

Bugs
Paarthurnax (Quest) Paarthurnax (Misión) Партурнакс (Квест) Paarthurnax (Missione)