Legal Question in Family Law in California

Yes, absent a court order otherwise, which your son may or may not be able to get. If the current order is for joint legal custody, any commercial use of the videos might require both parents, but just posting things for fun does not. If your son thinks this is in any way harmful to the kids, he should take it before a judge for an order that it stop, but absent some harm to or negative impact on the children, what each parent does in their own house with their own custodial time is their own decision. Personally I think it's a bad idea to publically post photos or videos of your kids on the internet, and my partner won't let me even share our son's photos on private secure sites, though I think that is a bit much. So a judge may agree, but you have to get the order before the ex would be doing anything illegal.