All centres I have seen have a big on the door of each room that says:

(insert number here) cases of (insert illness here) reported in this room in the last (insert number here) weeks.

Maybe suggest that your centre does something similar so that parents know if gastro/flu/hand,foot& mouth/conjunctivitis etc
Actually, they HAVE to do this. It's part of the NHMRC guidelines. They also have to have a folder where every reported case is documented, even if the child isn't at the centre while they have it.
If you want to be sneaky, phone the relevant State Department and report it anonymously. This will lead to a general spot check of the Centre (I promise that they can't possibly know it's you, spot checks happen all the time anyway) and they will get put on notice for it. They'll have to fix it - but won't get shut down or anything, there's enough of that happening around the place!

Good Luck and give 'em heaps!