If a son/daughter tells one parent something in confidence, and asks that the parent please not tell this item to the other parent, can the first parent keep the confidence or is the first parent obliged to tell the second parent?
Note that the parents have very different relationships with their sons/daughters.
It is hard to answer such a question without knowing the nature of what was said and its circumstances. In general though, when anyone tells you something in private we may not tell it over to others, because it is considered rechilus, and lashon hora to say another person’s secret to others. Even if it is a parent who asks us to tell them this secret.
However to reiterate, it is difficult to answer definitively, without knowing the situation.
Yuma 4b, Magen Avrohom O:CH 156