Oh Honey, you've got to put a stop to unpaid days for sure! I charge a set fee per month, which clearly states that this fee is set regardless of child's actually attendance due to them having a family vacation, sick days, relatives visiting, etc. The way you can explain it is to bring up examples such as paying rent, or tuition. It stays the same and is expected regardless of whether someone decides to camp in a tent for a month, or goes on vacation. When people pay you, it should be for the space that you are securing for them and their child for the month, regardless of the attendance.
And for part time, I always establish a set amount of days which have to be paid for each month as well, regardless of attendance. So one little girl I have, her mom is a flight attendant with an unpredictable schedule but said she would most likely need 10-12 days per month, so I told her I would take her in but the set fee would be for the 10 days no matter what, if there were days that she was not here, etc. She would add to the pay of course if she ended up using 12 days. And I also had to think about late charges due to her job and additional fees if she was staying and eating dinner with us, etc.
If people have a problem with that, then I wouldn't consider them secure enough, or fair enough to me to be my clients. Its the whole babysitting notion that is the problem. Some people think that if their child is not here, why should they pay? But don't know that a dayhome/daycare is a much bigger committment and responsibility than that.