If the parents are together, they have shared custody similar to divorced parents with that custody stated in writing. So they probably do need the permission of the other parent. But since they are together, most legal jurisdictions would assume that without some order preventing such travel, the consent of the absent parent is implied.
In other words, a parent can probably prevent such travel. But just not stating consent won't be enough. Some legal entity would have to be contacted somehow. It would only be a violation of the law if a parent crosses state lines in violation of a written custody order.
If the parent not wanting to give permission wants to avoid legal fees and court orders, they can mention how they COULD get an injunction and that COULD be the beginning of divorce/separation proceedings and how such travel prior to any proceedings that is not unilaterally agreed upon COULD make a judge think that "we" won't be able to effectively co-parent the child[ren]. And throw in the thought that judges might rule against a JW on such matters.
But seek legal advice. I just think in legal-eeze.