Major record companies have legal departments. They're not going to greenlight an album cover without rights to the images. Anyone who works in publishing knows how strictly this stuff is treated at a major label.
The record company should (only slightly ironically) keep records.
If they exist this case is going nowhere. If they don't - that might get interesting.
It's hard to imagine a situation where a record company ends up with naked pictures of someones kid, the parent gets paid, and the parent is genuinely not ok with the pics appearing on an album cover.
I agree it's more interesting if there's no consent form, but I still would hope the court would side with the record company.
The money was given for some reason. What reason did the parents think?
A written agreement definitely helps to clarify the matter, but is neither necessary nor sufficient to guarantee a resolution.
A verbal agreement absolutely can be an enforceable contract. The crux is whether there was a meeting of the minds. Of course, without documentation this is hard to prove, so you find people arguing it either direction in legal disputes.