If there is a posibility of something happening, given an eternity, it will.
We have allready seen paarth betray mortals before. We know it is possible.
And Paarth is still immortal, like all dragons really are (apart from when they are killed by a dragonborn). But just because something is immortal it does not neccecarily mean it will not age, Otherwise all dragons would really still be tiny whelps (asuming ofc that in the TES universe dragons weren't created as big as they were, I'm not too sure on that part). If that is the case then maybe Paarth was created to look like more of an elder. Or, there is allways the possibility he isn't old like we think it is. Those could just be battle scars we are seeing on him.
Expect that these dragon do age, which would make sense when one would consider that all the dragons that use to work for the Empire is gone right now. Heck, Paarthurnax is consider the Old One by other dragons and he "felt" young again when one return to the Throat after beating his brother in Valhalla. Then there another account that from the http://www.imperial-library.info/content/olaf-and-dragon that the Dragon is getting too old and might as well go out with a bang. There ins't really any support of them living "immortally" via age like the PC of Morrowind.
You're missing the point - if you sit on the fence, you cannot reasonably expect either side to take any risks on your behalf. And that's exactly what the Thalmor would be counting on. It's clear that the Empire is no threat to the Thalmor currently, and the smart thing for either side would be to consildate, get rid of some dragons first, and go after the Thalmor later. I wonder how much reverence the people of Skyrim would have for the Greybeards once word got around they're protecting one or more of them....
What the Thalmor is counting on is that the Civil War continue on for many more decades, which would give them enough time to rebuild their army to destroy the Empire. What they didn't expect is the PC getting involve with one or the other side. The unexpected end of the war would either force the Thalmor to create another plan to prolong the Empire's resource while building their or hope that if the Empire lose, they don't see this as a wake up call and the Empire start being isolated themselves to be stronger without of the Skyrim help.
As for the fence, the Greybeard would last in if either side is victorious. If they ever get involve with one or the other side, then the trust they have with either side is gone and if the side they join lose, they are gone. Beside, we are talking pacifist monks here; they don't usually go to war or try to get into politic, as mention a bunch of times. Its like being against their way of life.
@Bigbossbalrog: Agree'd. I salute to the last Blade, who died in honor. You would not be miss, Acilius Bolar.