Ok I think they believe you work for them. It is up to you to make and enforce the rules. My policy book says you pay for the day regardless of attendance. And it's the full day fee not a $10 charge. My part time rate is more then my full time rate and my part time is a min 3 days/week. My policy book also says I can make changes that I deem necessary with a two week notice for these changes to take effect. So if this were me I would hand them a new contract with a new rate at 3 days per week ...set days .... No changes and if they want to add a day then it's extra. At this point they are paying you $70/week (2 days at 30/day plus the 10$ for the third day they don't come) I think I would be okay if they went else where because they don't respect you or your business, and I could find a way to shave $70 a week off my expenses just to get my self respect back.
I would raise my rates and make part time even higher and set a minimum days attendance and give it to this family and all future families. Hopefully you already take a two week deposit to be used for the last two weeks of care so you don't get stiffed. Let this family try and find someone else to look after their child for that little bit of money.