I have learned over the last year and a half that in order to run this business properly, I must have what I like to think of as "rigid flexibility".
I have no wiggle room on hours, fees or our daytime schedule. If a family cannot work with those things, then they do not have a place here. I completely agree that very often, it comes down to priorities.
You need to take your business (and your income) seriously if you want parents to do the same. I get that sometimes we feel bad about people's situations. And I would never leave someone in the lurch with no childcare, if that could be prevented. But let me ask you this; if one of your parents got called into the boss' office and the boss said "hey, times are tough, s#@$ is going on, we need you to come in and work but not get paid as much".... chances are, they would be ticked, no? Or "we're not sure when or how often we need you, or how much we're going to pay you, but can you just hang around and wait for us to say"? That's essentially what they are telling you.
I would absolutely fill that spot full time. If you want to continue working with this family, go for it. But I would decide for myself; if it was ok for them to call the shots on what hours/days they need, or if you need to work out a loose schedule of sorts. I don't think it's too much to ask of them.