The only way I would do it, is that they understand that they will need to start paying the full-time rate or lose the spot if another full-time family comes along.
I know many people say don't bend your rules, and I agree, but I remember when I first opened, I made a lot of deals just to get families in the door.
The downside to this was I was not in control of my business in many ways, parents ran the show. But it kept the lights on and food on the table. As time went on I weeded out the bad apples and adjusted the hours, rules and fees for the families that I did want to keep.
Do what works for you, but just keep in mind that you may end up with much more stress along the way! Good luck!!