I think it may be more of a macaw thing and a cockatoo thing than a general parrot thing. I've seen lots of 'toos do it as well. As for my four macs, Zaf and Salsa do the up n down bobbing all the time. Whereas Jack hasn't ever done it during the nearly 36 years he's been with me! He does however do his side-to-side swaying.
So since Jack and Lucky are both B&Gs, this suggests to me that the bobbing-dance proneness may not be so much species-related as it may be personality-related. But I also think both of those aspects may contribute to some degree.
Then there's Maynard. He does neither the bobbing nor the swaying... so go figure . This all suggests that while the up n down bobbing seems more of a macaw and cockatoo thing than a general parrot thing, it as well seems to be directly related to a bird's particular personality. In any case, I think it's a cute and interesting trait