I wouldn’t be so sure about that. Courts have found that men were liable to pay “palimony” to ex-girlfriends, after the relationship ended, because the woman was able to make the case that there was a “promise of support”. So, if the kid can convince the court that the parents made some promise like that, an idiot judge may find them liable.
There would need to reliance as well, but that wouldn’t be too hard to prove. :-)