Okay, a couple points....
My contract stipulates that I can make amendments with 30 days notice. As in; if you are not accepting of the changes, you can give your notice in that time. I don't think that's unfair, as it goes both ways. If the parents change their situation/arrangement, etc, they give me the same notice.
I know it's too late now, but I don't think I would have "asked" the parents. I may have floated the suggestion, to see what they think/feel. But it's ultimately my (your!) decision, not theirs.
Anyway, if you feel like there is no reason for you NOT to take on a child, you need to discuss this with them. Find out what exactly makes them uncomfortable. Perhaps they don't realize the schedules, in that you won't have all the kids, all the time.
Ultimately, they can't DISALLOW you.... but they can leave, for breach of agreement, if they get right pissy. So, if they are still against it; you need to decide what means more to you. That family, or adding some more income?