It turns out there are at least a couple of dozen cases, from many jurisdictions, that talk about a noncustodial parent's "right of first refusal to babysit the children."

This makes perfect sense -- I assume it means that when the custodial parent wants to leave the children with a babysitter, the noncustodial parent should get to be that babysitter if he or she so wants, much like a right of first refusal operates in commercial contracts. But it just sounds a bit odd, given how commercial "rights of first refusal" usually are. I take it that when two parents can't stand to be near each other, and drop the children off at a third party, we'll be hearing about how the children are "in escrow."