I don't think issue has been explored in a systematic way, and it's not clear how it could be. Theoretically, one might record human language contrasts like tal, thal, ttal uttered by a parrot (how do you decide that the parrot intended to utter tal versus thal?), and present them to human speakers of the language, to see if (without training) they correctly identify intended "tal" utterances vs. "thal" vs "ttal". I assume that Georgian speakers can make sense of Chiko; if you edit it down to just one word so that you can't rely on context – and where phonetic precision of necessary to distinguish words of the language –
It is pretty clear that they have at best approximate skills in reproducing human sounds (they are kinda small, so it's physically impossible for them to sound like an actual human). This bird is pretty good; but focusing on the bird's actual output and not the owner's suggestions, it's a little less impressive. at 47 sec in there is a word which is potentially one member of a minimal pair (initial voicing) and I wondered whether I would perceive it as voiced or voiceless. As you can see, it doesn't actually sound like speech. You can compare "go" at 60 sec and "clover" at 78 sec, again looking for evidence of control over voicing. Of course, this is cute animal tricks on Youtube, not a controlled scientific experiment, but I think it indicates some of the problems with performing the experiment, especially in a way that the test was about the parrots and not the humans making judgments about the parrot sounds.