Right now I'm preparing to return to Slyrim for my final try at a build, and I decided for a Blades-centric one, as I believe they are THE go-to faction for a dragonslayer (also I already mentioned how I love the Akaviri aesthetic, as I'm a professed fan of Japanese culture, from which Akaviri culture draws heavilly from), and though in this build my character will also join the Imperial Legion and the Dawnguard, these questlines will be taken AFTER the Main Quest.

One thing I remembered as I worked on my build is how odd I always though that Delphine would ask my character to kill Paarthurnax for his past crimes, yet, she never considers Odahviing. At first I thought it was because Odahviing swore his alliance to the Dragonborn after he offs Alduin, but Nahfahlaar, the red dragon who perished in the Battle of Stros M'Kai during the Tiber Wars, had a history of allying with humanity.

The answer is in the Atlas of Dragons, a book that pops up in Sky Haven Temple after you defeat Alduin. This book was written by a member of the Dragonguard on 2E 373, and lists the dragons that were, at the time, deceased by report, slain by the Dragonborn, and known to live. This is the list:

DECEASED BY REPORT

Nahagliiv - Local tales name him as the dragon buried in the mound west of Rorikstead. No date associated with his death, although almost surely dates to the Dragon War era.

Odahviing - Records dating to the Crusade of interrogation of captured Dragon Cultists indicate that this dragon was buried in a mound in the southeast of Skyrim, near Riften.

Sahloknir - Local legends claim this is the dragon buried in the mound near Kyne's Grove, slain by the Nord hero Jorg Helmbolg in the First Era.

Viinturuth - Death dating back to Dragon War era, according to documents recovered from Dragon Cult temples which record his burial near Lake Yorgrim.

Vuljotnaak - Death dating back to Dragon War or just after, according to recovered Dragon Cult documents, which record his burial in a mound near Granite Hill.

SLAIN BY THE DRAGONGUARD

Grahkrindrog - Slain in 2E 184 after perpetrating great slaughter in Winterhold and Eastmarch. Name confirmed with assistance from the College mages.

Krahjotdaan - Slain in 1E 2871 in the southern Jerall Mountains, name confirmed by the dragon's own account.

Unnamed Dragons - Numbering 12, as recorded in the Annals dating back to the founding of Sky Haven Temple.

KNOWN TO LIVE

Ahbiilok - Sightings dating back to the early years of the Dragonguard throughout the northern Jerralls [sic]. Multiple attempts to kill him have failed. He is believed to be lairing somewhere in Morrowind.

Mirmulnir - Last sighted in the Reach in 2E 212.

Nahfahlaar - Repeated alliances with mortal protectors which have prevented his elimination. His last known protector was the King Casimir II of Wayrest, which the Dragonguard successfully ended in 2E 369. He escaped and current location is unknown.

Paarthurnax - The legendary lieutenant of Alduin in the Dragon War. He is now known to lair on the Throat of the World under the protection of the Greybeards of High Hrothgar. Master Araidh continues the established policy of avoiding direct confrontation with the Greybeards while waiting for an opportunity to exact justice upon him.

According to this list, Odahviing was slain before the Akaviri arrived in Tamriel, maybe even during the Dragon War, while both Nahfahlaar and Paarthurnax were spared, maybe both for rebelling against Alduin.

This explains why Delphine never asks you to off him. Odahviing was killed once, so when was brought back to life by Alduin, it was akin to a criminal getting sentenced for any number of time in prison, and when their sentence plays out, they are released with a clean slate. People might remember their past crimes, but they paid their dept to society.

Meanwhile, though he was the reason humanity was allowed to live past the Dragon War, Paarthurnax never really paid for his crimes, and instead, helped Jurgen Windcaller found the Greybeards.