Our parents only clean to prepare for a meet and help with clean up after a meet. It's one of the ways they can earn their volunteer hours through the Booster Club. On a daily basis, no way do we ask parents. Clean up is all divied between the staff on a daily basis.
As for cleaning mats. Bleach, bleach, bleach. We use approx 1/4 bleach, 3/4 water. No bleach on the beam, vault or carpet of course. You can spot bleach bars if there is a bloody spot from rips, but other than that Lysol spray. Lysol spray for anything we can't bleach. The floor carpet area is always our problem spot. Lots' of vacuumming weekly, but it's difficult to truly sanitize.
This season we keep Clorox wipes handy all around the gym to clean up the preschool drool/snot etc. as it happens. Gross, I know... but reality of the trade. We also don't hesitate to send nose pickers directly to the bathroom or hand sanitizers if we catch them pickin. We have kleenix all over the gym for them to use so we are trying to "change the habits" in the gym.