Depending on the species, the parrots may be a problem. Quakers, for example, are banned in many states as are Indian Ringnecks.
There's a list at http://www.fws.gov/injuriouswildlife/index.html of species covered under the Lacey Act but you will have to check each state individually for their restrictions.
I've heard that California bans ferrets and hedgehogs among others, and is pretty strict about enforcement.
I researched this issue myself because I have Java finches. Unfortunately they're listed under the Lacey Act so they can't even cross state lines.