Vanessa:
They dry themselves. Give the caique a folded towel to dry off on & watch the fun
This is how you entertain your friends now--LOL
The cockatiels will preen themselves dry. This all somes under the broad heading of enrichment. It gives them a natural behavior to do which is so important for their well being. So get everyone in the shower on perches then you get in there too & have a whistle fest.
We got a showerhead from Costco that has a very hard spray setting that I point at the shower wall to make a mist & that gently bounces back onto my caiques & they just love it. It also cools the water, so you can have warm water for you & they get cool. Do your showers in the day time.
Ingrid & the girls