To drink, most birds need to tilt their heads back and let gravity do the rest, but I know there are a few birds that can swallow.

My question, how do parrots drink? Most specifically, do they have the throat muscles necessary for them to drink without the aid of gravity?

Unlike Geese, and swans, parrots dont need " gravity". they have throat muscles to swallow. Proof in point- they dont need to lift their heads to eat unlike geese.
One time someone shipped young swans. However- the cage was too small for them to lift their heads all the way to drink. We had to let them out to drink.

Most parrot speceis do not need to tilt their head back to drink. My conures don't need to, they'll stick their beak in there and drink up. But my cockatiels always tilt their heads to drink.

yEA MY COCKATIELS DO THAT TOOO!

My cockatiels lift their head after they take a drink, but I don't think they are so much tipping their head back as just lifting it out of the water dish.

My lori uses the little hair like cillia on the end of his tongue to drink. I think it is neat to watch.

My lovebird tilts his head to drink... ( hes drinking again )