Concerning pet parakeets, Shabbat & Yom Tov, and the care of a domestic creature that is dependent upon humans for its care: If the birds are allowed to fly free around the room for a few hours every week day, and, on their own volition, return to their cage without any action on our part, then is it permissible to allow the birds out of the cage on Shabbat & Yom Tov, and to close and lock the cage door after they return to the cage? Is Shabbat different from Yom Tov in this regard?

It is permitted to open the door and lock it after the parakeets have returned on Shabbat and Yom Tov. (See Shulchan Aruch OH 316;12 and Mishna Brura there notes 53 and 57)