There are two different ways in which a parent can abduct the child.
The first is by taking the child to another state without the consent of the other parent. This is only the case if the other parent has (joint) custody over the child.
The other is by not returning the child to the home state, without the consent of the other parent. Again, the other parent must have (joint) custody, to make this a case of international child abduction. This kind of abduction can happen for instance after spending a holiday in another country.