If the parent insists that the child go in undies have them sign an agreement that they will pay for full carpet and furniture cleaning done by a professional company. Get a quote for an emergency clean from a company you believe can come out quickly and have the parents sign an agreement to pay and put a deposit of one cleaning down.
Once you attach money to the accidents they are usually willing to keep the cheapo diapers on the kid. Diapers are WAY cheaper than carpet cleaning unless you are not the one paying for the carpet cleaning. If someone else pays for it it is cheaper for the parent to do undies. Make wearing undies expensive and they will be more than happy to do diapers.