So I was wondering this the other day, while playing through the main quest again.
Do dragons get hungry?
I'm going to say yes. Mostly because at a few of the outdoors dragon shrines, there is often a dead cattle which appears to have been eaten.

So here's the proper question - how did Paarthunax survive all those years at the top of the Throat of the World? I would assume that the Greybeards provided him with food, but that is only an assumption. Would he go out hunting; but how is that possible, surely someone would have seen him.

It's just one of the more silly lore questions I guess, but I do wonder.

Paarthurnax can hunt for himself. He just have to be careful not to be seen and there are lots of space of Skyrim where nobody ever set foot.

They must eat, the kind of loot they drop includes armor and weapons. I've always assumed that these items are pulled from the Dragon's insides, kind of similar to the way really weird stuff turns up in the stomachs of big sharks.

I have no idea if they experience hunger, however.

I think the Nords use cattle as sacrifices/tributes for both giants and dragons.

I assumed he was able to go for longer periods of time without eating than other dragons by meditating like the other Greybeards do. They just get meager amounts of food from people delivering it up to them from Ivarstead, so I assume if a human can do it, a dragon probably can too.

I imagine that in a pinch, Dragons can survive on language alone.

Paarthurnax doesn't stay on top of the throat of the world permanently. He's said he used to visit the dragon caught in dragonsreach. And he probably isn't completely averse to hunting. If you do Froki's quest, he talks about Kyne and hunters.. she doesn't approve of mere butchers, but hunting can be something that treats life with respect.

Well as you mentioned with the dead cattle, and also the items found inside dragons suggest that they do eat, but whether they need to eat is different. I would say that since they are virtually immortal, barring being killed by another dragon (Or of course somebody with the soul of dragon), they likely do not experience hunger, and eat for pleasure rather than need.

EDIT:Fixed a typo, I should really start proof reading before I post rather than afterwards....