Why Parrots? Open question there, but I will answer what I think you mean. I see Parrots as I do Mockingbirds. Both mimics but with different diet and thus adaptive beaks. All songs serve the same dual purpose in birds. I think mimics are more evolved. Clearly higher processing is required to replicate other species' songs. We know, or it is believed, that mimic behavior is used to attract mates and communicate territorial claims. Further, it has been shown that Mockingbirds with the best repertoire of songs (esp males) have better breeding success than those more limited. So, I would assume the same would be true with Parrots- although I have studied Mockingbirds as a hobbyist but not Parrots. But in the same way mimic behavior and the higher intelligence required to produce it would be an adaptive advantage, right? ie, the female would select the best and smartest genes to pass along in response to this behavior.
What the female is looking for is some sign that the parrot is her kinda' guy ~ not that he has superior genes ~ just the same genes.
His ability to produce the signal that conveys that information is the part that tells her if he's capable of helping feed the chicks.
BTW, all the parrots of the same species have just about the same genes ~ maybe a few differences here and there with methylation ~ but same same!
My concern with parrots is they have more advanced brains ~ and yet they don't do very much different from other birds ~ in fact, my son prove that you can call a parrot down from a tree to sit on your shoulder just as you might a crow or a duck. Surprised his girlfriend I'll tell you that, but not me ~ or our other relatives.
Seriously, being a bird is not a mentally challenging occupation