OK first of all you need to revisit your contract ASAP!! You need to enforce a late fee for late payment. Also payment needs to be paid Friday at drop off!! I have no problem asking for a cheque. If they don't bring a cheque that Friday morning they cannot leave their child. I sure hope you are getting your pay 2 weeks ahead as well! My late pick up fee of $15 starts Friday at noon and continues everyday until payment is received. No payment or late payment is grounds for termination. You need a solid contract and need to stick to it.
I would tell all your parents that your spots are all full-time and if they want part-time hours that they are still required to pay full-time. Do you get paid for your stats? That should be another thing in your contract that all stats are paid at regular fee.
I would spend the weekend redoing your contract and hand it all to them Monday morning. You need to get strict and fast. Everyone must follow the same rules and do not EVER do anyone any favours.
My families wouldn't even ask to not pay, pay later or pay half etc. They would be out the door and I would never look back. If you would like to see my contract send me a message and I can email it to you.
Also quit texting everyone. Tell them to phone you and talk that way. Do not negotiate either. They should also be required to give you 2 weeks notice before pulling their child so the one who wants only b/a school care should be paying for 2 more weeks for the other child. Hopefully that is in your contract as well.
Good luck!