Does the parrot card have to be named "[something] Parrot"?
Yes.
Parrot (Just Resting) Parrot (Stunned) Parrot (Pining for the Fjords) Parrot (Prefers keeping on its back! Remarkable bird, innit? Lovely plumage!) "And so on and so on and so on."
As I've said, one of the things I love about this card is the way the contradiction between the picture and the words kick-start the whole argument which the card is having with itself. The person who plays it instantly takes on the shopkeeper's role, insisting that the bird is alive, and pointing to the title as proof. No matter how much the other players may act like the outraged customer, pointing to the picture and saying "that's a dead parrot," the player with the parrot automatically counters with claims of the parrot simply being unconscious. This all becomes another reason in my mind to call it a Sleeping Parrot -- precisely because the word isn't used in the sketch! Since there's obvious fun to be had in remembering for yourself all the different lines from the sketch, and firing back those quotes, isn't it better not to waste any of that material?
Calling it a Sleeping Parrot gives the quote-master one more line to feel clever about using, while calling it a Resting Parrot takes away one chance for players who really know their stuff to prove it.
Also of course, I really liked what Carl said:
But putting "sleeping parrot" on the card doesn't harm the memory of the sketch at all. "Sleeping parrot" plus the drawing of a clearly dead parrot captures the actual humor of the sketch on a card better than any other phrase, no matter how "authentic" that phrase might be when compared with a transcription of the sketch. In fact, it's the only suggestion so far that sets up any humor for someone entirely unfamiliar with the sketch.
Exactly. So I'm back to thinking I should stick with Sleeping.