Parrots and other such birds are believable to begin with since we know they can talk in real life. In fact, it's something of a relief. Second is something called personification. A bird with a peg leg or a cigar may seem like a stretch of the imagination, but it actually isn't. These are classic examples of personification.
People personify pets all the time.
For example, we project our own human qualities onto our birds when we interpret what they're thinking or feeling. When Disney's robot birds talk, it's easy to forget that the words are from a script. Anyone who has attempted to train a bird to talk knows how tough it can be. Birds naturally watch and respond to the world around them, but though they use our terms, they do so on their terms. Throughout the ride, parrots mock their pirate masters as they pillage a Spanish town.