Indeed, that is true. I'm sure they would adjust in an aviary, once they realize they're getting free food and kept safe from predators.
You would need some strong material for a cockatoo aviary though, or else they would chew right through it.
I would figure that would be the case for macaws and other large parrots.
There has been parrot species that have suffered extinction, through both shooting and captive breeding. The Spix macaw officially became extinct as a wild species, and since there isn't enough blood diveristy in the captive birds that are still alive, many babies end up with many inbred problems. If people were to transport wild caught birds from australia to america, one would at least have to be mindful of at least keeping a number wild, lest they extinct the species.
The Carolina parakeet suffered extinction, due to farmers shooting them for eating their crops. They were also bred in captivity, though no one was willing to try and save the species as a whole, at the time. In result, the U.S. lost one of two native parrot species (there's still the endangered thick billed parrot, which only lives in mexico now, and there are attempts to try and reintroduce it to the U.S.).
Though, there is still the invasive quaker population here, in which people are debating catching and using for pet trade.