Paarthurnax (Quest)

Background
Delphine and Esbern have discovered that Paarthurnax was one of Alduin's chief allies during the ancient dragon war. They will refuse to give any more help until you kill Paarthurnax.

Overview
This quest is purely optional. It's best to perform it before completing the main questline if one wants to continue Blades storyline. However, doing so will prevent any further communication with the Greybeards. NOTE: If you choose to kill Paarthurnax, the Greybeards will stop pointing you towards new thu-ums.

Trivia

 * Should the Civil War questline be complete (thus bypassing Season Unending) and the player proceed directly to the endgame, Delphine and Esbern will remain in Sky Haven Temple, however, they remain in disapproval and the quest disappears from your log.

How NOT to kill Paarthurnax (PC only)
It is possible to complete this quest without needing to slay Paarthurnax by using command: setstage MQPaarthurnax 100 This way, Paarthurnax will remain alive, the Greybeards will still be hospitable and the Blades will talk to you once more.

There are also mods available to bypass this quest or the player could use the console command (setessential  0) and kill the blades instead...

Bugs

 * After beating the main quest Paarthurnax becomes unkillable. However, some players have reported that attacking and killing him will complete the quest, even though it has disappeared from the quest log. (Confirmed) However in some cases Paarthurnax will remain unkillable even after the quest has disappeared, preventing further progress along the Blades storyline.
 * There is a way for you to kill him after you have killed Alduin, if you use Dragonrend to make him get off his pedestal, you can then kill him when he lands(confirmed)
 * After the main storyline is complete you can still travel to the Throat of The World and attack him after awhile he'll become hostile fly up and attempt to kill you after you kill him you can finish the quest.(confirmed)
 * Confirmed on PC, using storm call on Paarthurnax made him fly up & subsequently attack. Although quest to kill him was not viewable in journal, update did occur after his death.
 * On Xbox and PS3, there is a glitch where Paarthurnax is unkillable; directly after receiving the quest. However this glitch allows you to level up your skills.
 * Save before you try attacking him, in case the glitch where he becomes invincible takes place. If it does, simply reload your save and retry. It also helps if you do not engage in conversation before you kill him.
 * One way to ignore the glitch where you cannot engage Paarthurnax is to get your follower to attack him. Paarthurnax will fly up and get into an attack position; from there you can kill him. (Confirmed on PS3, 360, and PC.)
 * Follower says that attacking Paarthurnax is "not possible" so the above option may not work. Instead, use Dragonrend and attack a few times. Your follower will join in and Paarthurnax will become hostile.
 * Another way to bypass this glitch: do not use Dragonrend on Paarthurnax, let him fly off. If you try to kill him while he is perched on the wall, he will be unkillable and not fight back. If he doesn't fly off, try using ranged attacks (magic may work more often).
 * Another possible way to override this glitch is to attack Paarthurnax with Mehrune's Razor (possibly due to the instant kill enchantment on the blade) which entices him to reciprocate combat and attack you.

Another way of killing Alduin & Paarthurnax:
 * Once you have returned to the Throat of The World after defeating Alduin, speak with Paarthurnax. After a brief dialogue, he will begin to fly away; once he is hovering in the air saying his final farewells, attack him a couple times and he will become a killable target and try to fight you. This method allows you to complete the main quest line and also continue the Blades quest line at the same time.
 * After the first fight with Alduin, you must speak with Paarthurnax; after initiating dialogue with him, he may randomly fly into the air and spin in circles while diologue remains active. He will get too high and you won't know what he is saying without subtitles. This problem can be fixed by exiting dialogue and coming back to Paarthurnax later. (Confirmed on PC, Xbox 360, and PS3)
 * After you have finished the main dragon questline, you will most likely still encounter this glitch although his dialogue will vanish.

Order your follower to attack Paarthurnax and then he will fight back.
 * After killing Alduin, I couldn't kill Paarthurnax until after I had talked to the Greybeards one last time and exhausted all dialog. I then went to The Throat of the World and used Storm Call and a battle axe and finally Paarthurnax flew up and then fell to his death and even though the quest was no longer listed I was able to finish it. (Xbox 360).
 * If you finish the civil war quests and skip seasons unending you must go to Sky Haven Temple and talk to Esbern or Delphine for this quest to trigger. (DO NOT think this is a way to be able to talk to both blades and greybeards forever; the second you walk into Sky Haven Temple after The Fallen you are not a member and taking items is theft.)