Auggie's is usually just room temp or even cooler: cool tap water.
I know many people recommend it being warmer, but he only goes for cool water.
More importantly though is the air temp after the bath when they are drying. A wet bird coming from any temp water into a cold room will get chilled much more than a bird who took a cold bath and is drying in a reasonably warm room.
it's two different temps for my girls ... my GCC like ICE COLD water and my sunny (the spoiled rotten princess) won't get in unless it's luke-warm ... there is an old saying for those that recommend warm bathwater for parrots, "no one heats the rain" so I don't think anything more than room tempature is necessary ...
Also, like AD said, watch the room temp after they are done ...
My parrot hates cold water.. - so she has a shower with me. I make sure i turn the temp down a little - so it's still relativley warm. My bird is an Eclectus - so they come from the humid parts of the world. (Rain forests - up in the canopies). She is spoilt. She loves having showers - she nestles up to me and rests her head on my shoulder - just like a baby - and makes her cute baby noises and clicks her beak. Happy little vegemite she is.