All of my larger parrots withdraw a foot into their bodies and perch one-footed. They tend to do this most often at night, when they are a little sleepy. I believe they sleep one-footed as well, but, believe it or not, I have never caught them sleeping. If we're around they instantly pop to attention.
Usually they ball their foot up and bury it in their feathers, appearing to be one-legged. I searched for a picture of this, but don't have any. That pose must not be exciting enough for me to capture! I'll try in the future, but when they see the camera, they usually start strutting around.
This morning, Max was perching one-legged, in the position we call "The Claw." She is the only one of our parrots to do this. We're not sure why, but it makes us laugh. Thomas and I make jokes about the claw coming to get us and occasionally make claw motions at each other.
This next picture turned out a little blurry. I think she was unfurling her foot and making the begging motion to get me to pick her up, but it looks a little threatening when I captured it in this stage:Max says, "I'm going to get you!"