It's not really justice. It's revenge, and it's short-sighted. The crimes Paarthurnax commited were done literally ages ago, long outside mortal living memory. Yes, he did horrible things, and very likely deserves to die for them, but not a single one of the people clamoring for his death has any relation to those crimes other than reading about them in an obscure history book. Esbern and Delphine's outrage rings kinda false when you consider this:
Over virtually the entirety of Skyrim's recorded history, Paarthurnax has forgone his urges and remained atop the Throat of the World, peacefully guiding an order of non-interventionist monks and keeping watch against the eventuality of Alduin's return. One could consider it to be imprisonment for his crimes, even if it was self-inflicted. He not only turned against Alduin, but was the only reason the mortals won, either time. Kynareth and Akatosh may be responsible for gifting the Thu'um to mortals and creating the Dragonborn, but Paarthurnax was the first one to teach them how to access and use that power against the dragons. When you've never been victimized by him or even met anyone who has, killing him over ancient reports smacks of ingratitude.
Then there's the fact that Paarthurnax's conviction in the Way of the Voice can hardly be questioned. He's followed it for thousands of years. After Alduin's defeat, he tries to spread this philosophy among other dragons. Even if he's not universally successful (as Odahving predicts), any dragon that does convert is one less that's gonna be ravaging the countryside. Let him achieve his Ambition in becoming a Cruel Overlord by forcing dragons to deny their inborn natures to dominate and destroy.
And, as one last point to consider, 99 percent of the Dragonborns that are presented with this choice are hardly innocent, noble, pure heroes of righeousness themselves. Especially if they've done some of the Daedric quests.
