My answer to this is always "what would the children do if the natural parents were still living together?" The answer is that the child and parent would HAVE to sort out their differences because they would see each other every day. This might involve constant dispute and arguing but more likely, it would involve a small disagreement and both parties working out a resolution. The child would learn that confrontation is not always a terrible thing, it doesn't always end in disaster but it's a way of communicating.
It is VERY common for teens to rebel against a parent, to refuse to obey house rules and be disrespectful. It happens because as a child grows he/she realises that his/her parents aren't perfect and have flaws in their personalities. It can come as quite a shock for many kids and they often become angry with that parent.
In a "together" household, most probably the "favoured" parent would help to sort out a solution for both the child and the other parent, to enable harmony in the home. There is no such pay off in a separated family, and in fact, it oftens pleases the other parent to see the child turn against someone who they themselves dislike so much. It makes them feel supported and hence fills THEIR need for acknowledgement on how difficult they find their ex partner.
I have experience from both sides. I have 2 resident children in my house, neither of which want to see their respective NR parents, both children are made and encouraged to go, both children have found their own coping strategies through talking it through and working things out for themselves. They are both 15. I actively dislike both NR parents and get very angry and frustrated by their behaviours but ultimately I will not be able to control their behaviour now or in the future and enabling the children to cope with their other parents is something I feel is my absolute duty. Both are doing well in school and achieving as well or better than predicted, have friends and although are not perfect, are functioning reasonably well as teenagers.
We are also have a non resident child as part of our family who refuses to see us and have anything to do with us or our extended families. He is 14 and after being actively supported in his wish to have nothing to do with his NR parent is now refusing school, has welfare officers involved, is attending school counselling (on recommendation from school), has dropped social activities and is suffering from low self esteem. He was predicted all A stars in GCSE but is now not on target. Part of self esteem is feeling "part of something" and for a child to choose not to be part of such a large chunk of his and his siblings lives has understandably taken it's toll. His resident parent has refused to encourage him to attend family counselling or mediation.
A parent who actively refuses point blank to see their NR child, is rare in my experience and often will just need enough rope to hang themselves but ultimately you need the child to learn from this, to protect them might feel a natural, caring thing to do, but ultimately part of growing up is learning to cope with all sorts of situations. We often teach our child the practicalities of growing up, doing chores etc but just as important is it to teach a child the emotional confidence of coping with difficult people, even if that person, is your parent.