The correct choice between "Who's" and "Whose" in your sentence depends on what you're trying to convey.
"Who's" is a contraction for "who is" or "who has." You could use it in a sentence like, "Who's going to take care of the parrot?"
"Whose" is a possessive form that indicates ownership. In your sentence, "Whose parrot is this," you are asking about the owner of the parrot.
So, if you want to ask about ownership, the correct form is "Whose parrot is this?"