The following is speculation guided by common sense, since I have no background in this field:

Birds (not sure about parrots specifically) have tremendous abilities to analyze and deal with sounds (see also: cetaceans). You don't have birdsongs that sound as complex as many are if some kind of information wasn't tied up in there somewhere (usually, the information is along the lines of "this is my tree. You want your ass kicked? Where are the women at?")

I suspect that parrots wouldn't have all that hardware and software for mimicking noises if they didn't have a strong ability to respond to stimuli and respond with specific sounds.

If such birds were really able to use language with intelligence, they should be able to pick up communication skills from simply being around a human all their lives. A human infant does. Nobody has to spend all day "training" a baby to use language intelligently. Although parents help them out with words, for the most part they suck it up from people talking to them. My sister didn't talk almost at all until she was three, and her first real words were "I have to go to the bathroom."

So why can't a parrot do this? At some point, a pirate who had owned a bird for twenty years, looking for where he left his bag of gold, should hear from across the room "It fell behind the table, ya swab."