I'm no expert, but it seems to me that this is a significant discovery:
i) Lately, a number of professing Christians have abandoned the historical Adam based on (alleged) genetic evidence for the common ancestry of apes and humans. The basic argument is that common genes imply common ancestry. The more two creatures are genetically alike, the greater their affinity by common descent. They share so much genetic material in common because they share common ancestry (i.e. most recent common ancestor).
ii) However, even Darwinians admit that the shared vocal ability (i.e. imitation, vocalization) of parrots and humans isn't due to a common ancestor. Rather, they chalk that up to independent development of the same morphology and brain circuitry.
iii) But a familiar problem that convergent evolution poses for evolutionary theory in general is that it undermines evidence for common ancestry.
iv) Another issue is why we'd attribute this to independent evolution rather than special creation.
v) Finally, if parrots and humans have similar vocal abilities because they have similar genes, then how do shared genes count as evidence for evolution (whether convergent evolution or common ancestry)?
From a theistic standpoint, if God chose to make two creatures with similar vocal abilities, he does that by giving them similar genes. They have similar genes, not because they are related to each other, or because they evolved, but because that's the natural mechanism to produce a similar result. God uses similar physical causes to produce similar physical effects.
To say they are related because they share common genes is circular. You can reason from similar morphology to similar genes or vice versa. Each implies the other.
If God designed to creatures to have similar abilities, then the way to engineer that would be to give them similar genes.
Some creatures are more alike because God chose to make them more alike. Some creatures are less alike because God chose to make them less alike. The range of diversity is a tribute to divine ingenuity. Like origami. All the different figures you can shape by folding the same piece of paper different ways. That takes great imagination, artistry, and skill.