...But not by the hand of the Dragonborn. What Esbern said about his past deeds was right. Think about it:
St. Paul used to be a massive persecutor back in the early days of Christians. He then turned to become a Christian in his Epistles. However, due to his past sins, he still had to pay them by death nonetheless.
Paarthurnax is a dragon, the only inherit creature to be fully immune. He therefore, cannot die unless slain. Should he be justified, it should not be by the Dragonborn. But as much as I don't want him dead, he can't just dust off his sins like humans can. Since old age doesn't exist for dragons, he should die by some random avalanche or something other as weird... what?