You do not need a permit to destroy unprotected native animals by shooting. Unprotected animals are listed here. However you do need a permit where the destruction of galahs and little corellas involves trapping and carbon dioxide narcosis.
In any circumstances, when destroying an animal you must aim to minimise suffering of the animal. You must also comply with animal welfare standards outlined in the Animal Welfare Act 1985, the regulations under that Act and Codes of practice where they exist.