Costa Rican laws, which are unlikely to change, prohibit the importation of virtually all non-native species. That applies to not only birds and other live animals but also plants, seeds, bulbs, etc. The reason is clear enough. Non-native species pose serious threats to native ones. Absent the predators that control native species, on-native ones can thrive and crowd out native ones. These prohibitions do not apply to domestic dogs and cats.
Consider the problems posed by non-native rabbits in Australia and by non-native snakes in the Florida everglades. Google purple loosestrife in the U.S. Aspecies of non-native carp is virtually eradicating the native fish in the Potomac River.
Too, its illegal in Costa Rica to keep any wild animal captive except for licensed zoos, rescue centers, etc.