I wouldn't get too slap happy before checking with FWC. Like most states they do have regulations in place for non game birds. And like anything GOV they are mostly double talk... Some tidbits I found on FWC's website:
No one shall take, poison, store, buy, sell, possess or
wantonly or willfully waste wildlife, unless specifically permitted or authorized to do so.
Notice how that doesn't specify native or introduced or both. It is prohibited to intentionally place food or garbage, allowing the placement of food or garbage, or offering food or garbage in such a manner that it attracts black bears, foxes or raccoons and in a manner that is likely to create or creates a public nuisance.
Birds: State and federal laws protect native species of birds. Native birds may not be taken or harassed, except as described in these rules.
Sparrows, rock doves (pigeons) and starlings are not native. Migratory nongame birds that cause damage to trees, crops, livestock or wildlife, or that are concentrated in such numbers that they are nuisance,
may be taken with permits issued by the U.S. Fish and Wildlife Service by calling 404-679-7070. Blackbirds, grackles, cowbirds and
crows can be taken without permits when they are causing damage.
None of the species you mentioned are migratory as per se... they are all established there as introduced residents. Unprotected birds: State and federal wildlife laws do not protect non-native birds that have become established in Florida by accidental releases or unauthorized stockings.
Unprotected species include English sparrows, European starlings, rock doves (pigeons) and Muscovy ducks. If you are uncertain whether a bird is native to Florida, contact an FWC regional office.
Being introduced, parrots are not protected in FL, but as an ex parrot breeder and someone who has had them as pets almost all his life.... nada nada please. My buddy Bandit and I would appreciate it. 
Instead find out if you can get a permit to legally trap them... could add up to some big dollars in the pet market. FWIW
