Hello Don,
Yes Cockatoos are known to do this sort of thing sometimes. Or that is to say Galahs and Sulphur-crested Cockatoos sometimes indulge in chasing or shadowing in flight of raptors such as Brown Goshawks and Australian Hobbies. Probably other people will tell you observations about other species that do this. Of course many parrots are prey to many raptors. I don’t know that the parrots that harass raptors do it to the species combinations that they may be at danger from. It does suggest to me though that Galahs who often do this to Australian Hobbies and it as far as I know not with Peregrine Falcons, that are a major predator, that are able to distinguish the two species. Although I may be quite wrong and that may just be that this is what I have seen or read about.
You have a curious renaming there. “Glossy-black Cockatoo” would totally change the character of the name and suggested affinities of the bird. That spelling would indicate that the bird is closer to the other (white) cockatoos and not a member of the Black-Cockatoo group. It would suggest the black is described as glossy. The actual name is a Glossy Black-Cockatoo, which indicates it is a member of the Black-Cockatoo group. Although how it became known as Glossy is a mystery to me.
Philip