a parent is liable for their child's bills. Each parent would be jointly and severally liable to the provider. That means the provider could seek payment from either parent or both. If one or the other is liable due to a court order concerning custody and support, that means the parents need to deal with each other for who ultimately is liable but that does not involve the provider. That is between just the parents.
So, dad should pay the bill and if the court order is that mom pays, then he can go to court to seek and order for her to repay him. They have no business trying to get a provider in the middle of their arguments.

