My views are that as long as the animal's health isn't adversely affected, anything goes. And yes, that does apply to all species, not just birds. Many a mutation originally occurred in the the wild, after all.
Animals in the pet trade are virtually exclusively bred for the pet trade nowadays anyway, they're typically many generations removed from wild stock. (With some exceptions. But again, as long as wild-caught animals are sourced responsibly and ethically, with the ultimate aim being to produce captive-bred stock to meet demand for the species in the pet trade, so that animals will no longer have to be taken out of the wild to supply it, than even that is fine.)