You get the quest to kill Paarthurnax when you go talk to Esbern after completing the quest "Alduin's Bane". Esbern found out who Paarthurnax is, One of Alduin's chief lieutenant in ancient times, and responsible for many of the Atrocities that happened during the dragon war. And they want you to kill him. If you do not, they will not offer you any more help, effectively locking you out of Blades faction quests.

But Paarthurnax did help you and has seemed to change his ways, and is now a good dragon not an evil dragon. But when you try to tell that to Esbern and Delphine, your pleas fall on deaf ears.

Killing Paarthurnax is OPTIONAL. You might feel that he has helped you a lot against Alduin, and is no longer a bad dragon, so he does not deserve to die. Or you may agree with Esbern's point that Paarthurnax has committed atrocities against humans in ancient times, and only became good to save himself, but now needs to pay for his crimes.

Other then your conscious, you have to figure out which faction you want to side with. Siding with one faction effectively blocks out the other. If you side with the Blades and go kill Paarthurnax, the Greybeards will not be very happy. They will no longer give you assistance or faction quests. On the other hand, refuse to kill Paarthurnax and the Blades will shut you out. Esbern and Delphine will no longer offer you any aid, and you won't be able to start additional Blades faction  quests.

If you decide to let Paarthurnax live, this quest ends here and now. Just go about with your other missions and ignore this one.

Otherwise, should you decide to kill Paarthurnax, open your quest journal:
Delphine and Esbern have discovered that Paarthurnax was one of Alduin's chief allies during the ancient Dragon War. They refuse to give me any more help until I kill Paarthurnax

Kill Paarthurnax

Fast travel to Paarthurnax's home, the Throat of the World. Kill Paarthurnax, as Delphine and Esbern requested. The best strategy is to start with a Dragonrend shout, then attack him with all you've got, and keep shouting Dragonrend at every moment possible to incapacitate him

Once you have slain Paarthurnax, return to Delphine or Esbern at the Sky Haven Temple. Tell them it's done, Paarthurnax is dead. They will be happy to have you on their side. Esbern will be happy to bless you with the Dragonslayer's Blessing that gives you 10% increased Critical Hit Chance vs Dragons for 5 days whenever you talk to him. You will also be able to get Faction quests from the Blades.

If you travel to High Hrothgar to talk to the Greybeards, you will find out Arngeir is very angry. Paarthurnax's murder is beyond his realm of forgiveness. The Greybeards will no longer talk to you, give you quests, or assist you in any manner.

This completes the quest "Paarthurnax"