That link that you posted at https://www.plannedparrothood.com/sh.../shipping.html
specifically has the page title "Air Travel Guide to Shipping Birds". The page specifically mentions that shipping birds other than chickens, turkeys, guinea fowl, doves, pigeons, pheasants, partridges, quail, ducks, geese, and swans - is illegal. One can ship day old or adult birds of those species only. The Lacey Act is a 1900 United States law that bans trafficking in illegal wildlife.
A responsible pet bird breeder does not ship through USPS.