Eeek....makes me itchy just thinking about it.
@monotwine covered it all very nicely.
Another thought is that mites are generally quite small; most are too small to visualize unless they've been feeding, which makes them red. But you said yours were white, so I imagine they must have been a good size for them to be visible in a water bowl.
Lice are generally a bit bigger and more obvious.
My guess would be that someone bathed in the water and the bugs came off.
There are some good photos of red mites here:
http://birdmites.org/FAQ.html
They can be white/clear and then turn reddish after they bite the host. They're also nocturnal, I believe?
You could also check for feather mites quite easily. Check on the underside of the wing and look at the primary feathers for a "moldy" appearance, like this:
http://4.bp.blogspot.com/-F4dGksd5ed4/T ... +Mites.jpg
Or this:
http://aboutcockatiels.com/wp-content/u ... -shown.jpg
Or hold up a recently-shed flight feather to the light and you'll see something like this:
http://ts3.mm.bing.net/th?id=HN.6080241 ... d=15.1&P=0
If you can identify the parasite, you may be more effective in treating it.
I would definitely treat the flock and disinfect the aviary. I would do a second application of whatever product you choose. (Just make note of how long it's effective. If it's good for 21 days, then you'd want to do the second application after 21 days.) That will help ensure that it's killed off completely.