Did you beat Alduin in Sovngarde, or just at the Throat of the World. Until Alduin dies in Sovngarde, Parthurnaxx can't be killed.
Yes he can: if you use the Blades to get info rather than the Greybeards, then as soon as the Blades ask for his death, he can be killed.
However...
This does seem a little bugged on PC at least. The first time i tried he stayed set as essential. I reloaded the game and tried again, and this time he fought and died. One difference: the second time i didn't try talking to him first, just swung.