It would probably be easier to ask which species ARE permitted!
Basically, the Queensland state government seems to be hostile to any exotic species at all - well, maybe "hostile" is a bit unfair.
I just get the impression that most state governments in Australia would be happy with no exotics whatever. Queensland is stricter than most because it is mostly tropical, which means that escapees are not going to die out from the cold as they might in more southern states (I'm thinking mainly reptiles and tropical fish here); also, Queensland is heavily rural and the last thing that farmers would welcome would be packs of escaped hyenas running around, or hordes of ravenous coatis. (I'm being a bit sarcastic here.)
While most thinking people agree that there should not be amateurish private ownership of potentially destructive species, Australian governments, both state and federal, should recognise that professional, well-run zoos and animal parks are trustworthy and should be allowed to keep more exotics than are currently allowed.