The parent did get the best deal in the sense that just because you don't issue them a receipt if they have their cancelled cheques or some other proof that they paid you the lower fee they are probably claiming the amount on their taxes anyways.

It is also possible that they are being audited and need an official receipt to prove their expenses.

Not sure you will get anywhere asking for the back pay and may have to just go without that but moving forward from the date you give the receipt the parent will be expected to pay the same rate as the other families. I would give them a new contract to sign with the new rate and chances are that will get the truth out of them. Don't be surprised if they are planning to leave care and just getting all of their things in place first.