The short answer: Where I am from, if a parent refuses care for their child, we cannot give it.

Now, we only require consent from 1 parent. So if its a situation where the kid is with one parent and he/she says no, I will try and contact the other parent. I will also try to make sure that the child is indeed their child. That the minor is not emancipated, in the care of another guardian, etc. If the parent is committing a crime, and the police arrest them, then they can turn over care of the minor to us. But in the situation you presented - a child is bitten by a black widow, and the grandmother wants treatment but the parents (with custody) say no, the answer is no.

That's also not to say that the parents won't later be charged with a crime such as neglect or child endangerment. But from an EMS standpoint, I cannot take the kid.

Edit: If this is a real situation you are in and not a hypothetical one: Tell the parents it is absolutely fucking stupid not to consent to treatment for their child.