Paarthurnax (Quest)

Background
Delphine and Esbern have learned of Paarthurnax's position as leader of the Greybeards. They inform the Dragonborn of Paarthurnax's role as Alduin's second-in-command during the Dragon War, and his responsibility for the atrocities committed. Bound by their oaths as Blades, they cannot offer 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 purely optional. In order to continue 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. Players who decide not to kill Paarthurnax, however, receive a different conclusion to the main storyline.

Trivia

 * 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. You are also given the option to tell Arngeir, Paarthurnax must die or that you are going to spare him. This will affect the conversation between Esbern and Arngeir if in Season Unending, you are forced to negotiate a treaty.
 * 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.
 * It is possible to use the Greybeards for the Word of Power, then kill Paarthurnax and join the Blades to hunt dragons.

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 invincible. 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 invincible even after the quest has disappeared, preventing further progress along the Blades storyline.
 * Paarthurnax is essential while perched. The key is to get him off the perch. If you attack him while perched and if he loses too much health, he will not leave (you also have to be detectable (don't sneak). Attack him with paralyze (he cannot be paralyzed and your skill will not advance) and he will leave: this will do no damage to his health. As soon as he leaves, he becomes hostile and is killable. Dragonrend also does no damage but takes much longer than paralyze. (This is from the PC, but should be applicable for all platforms because there is no bug -- to be confirmed on other platforms.) Glitch still exists for his health being down to advance certain skills.
 * There is a way for him to be defeated after defeating Alduin, if Dragonrend is used to remove him from his pedestal, he can be killed upon landing. (confirmed)
 * After the main storyline is complete, you can still travel to the Throat of The World and attack him. After a while he'll become hostile. After you kill him you can finish the quest. (confirmed)
 * Confirmed on PC and Xbox, 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. (If his health is too low or if you are not detectable, this will not work.)
 * On Xbox and PS3, there is a glitch where Paarthurnax is invincible; 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 invinvible 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 that seems to work is to either use the storm call shout and knock Paarthurnax's health down or just simply attack him, and then fast travel somewhere. You may get a notification that Paarthurnax is dead and to report back to Delphine after fast traveling.

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 dialogue 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.
 * 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.
 * 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.)