Sure, depends on money involved, casino's reputation — if it's not shady, I would not mind handing over my kyc data. Though, child must have registered account by his person info, would casino allow parents kyc when details don't match?
...And if you do that, does it also mean that you are supporting underage gambling?
No giving my kyc would not mean I support underage gambling, I will withdraw the amount and do whatever is in my hands to make sure my child doesn't gamble again, at least until he's of legal age and on his own.