And yet, Paarthurnax was protected by emperors. So what? They listen and protect the Emperor only when they feel like it?
Also, for me, Delphine and Esbern are not "The Blades". They are members of that organization, not the Organization itself.
Quite simple and ill explain.
The emporers if they gave the ok to kill paar, they woulda had to go thru the greybeards first. Weither they actually got past the greybeards or got shouted apart, the people of skyrim would have been unfronted and most likely taken up arms as it was not only a grp they highly honored but also a declaration of war. The emporers would have stuck witha political fiasco and so they had a logically reason for not wanting the blades to go after him.
The dragonborn is neither emporer nor would suffer any king of fiasco or public outrage in killing paar since the dragonborn can walk freely up to paar. There is no political or public fiasco the dragonborn would suffer, only the displeasure of the greybeards.
So basically if the emporers would have given the order, they would be at war with skyrim. The dragonborn does it, he only has the greybeards pissed off at him.
Thats why. Nothing about paar being honorable or anything like that, it was at the time the action did not outweigh the consequences.