It all depends on the context, and what you are trying to emphasise.
If her previous green parrot was destructive as well, then the emphasis is on the fact that this is a new destructive green parrot. If you wish to emphasise that her new parrot is destructive, then it's "her destructive new parrot".
The same goes for the South American student. Whichever quality you wish to emphasise comes first.
I have to say that, as a native BrE speaker, I have never come across the list of "Determiners, Quality or Opinion, Size, Age, Shape, Color, Origin, Material + the noun".