Paarthurnax is completely optional. In order to continue the Blades storyline, players must complete this quest. If Paarthurnax still lives after Dragonslayer, the quest will vanish from the quest journal, but if he is killed it will reappear. If you want to speak about Alduin's defeat with Paarthurnax and the Greybeards, and join the Blades too, it is recommended to kill Paarthurnax after Alduin.