I'm confused by the whole situation. It sounds like you live together and they pay part of the expenses (food, utilities and some form of rent or mortgage) and you pay part of the expenses (rent). So who lives with who? Yes, this make a difference. What is the rent/living agreement not taking daycare into account?
Based upon the information you've provided, it seems like they are taking major advantage of the situation. How much do you make working 2 days a month? They certainly can aford to pay you for those 2 days, plus a little extra.
You are 100% correct, if they can pay a stranger, they can pay you. They have no idea how good they have it. I don't understand why you would be upset that they are looking for other day care. The way I see it, by doing the research they will see how lucky they are to have you and will appreciate more or they will put their children in day care and it will free you up to do your side job.
What makes you think they need financial assistance to pay for day care? They are both working. It's their responsibility to support their children, not yours.