After completing the main quest, the quest to kill Paarthurnax disappeared from my active quest list.
Any fixes for this?

Yes, go back and kill Paarth before you finish the MQ.

Same thing happened to me on my nord character, and I killed him as soon as I was returned to the throat of the world. That was a fun battle, because my Dragonrend so happened to hit three other dragons in the area ;D

Not sure if this is a bug at all now. I just went and killed Paarthurnax with no issues what so ever and the quest was recognized by Delphine and the game acknowledged it as well.

Just kill Paarthurnax earlier so you can ride on Delphine's back to save the world from destruction.

You people make me sick...