From the viewpoint of a "traditional household" that's exactly how they view their kids, and that's what they really mean by parents' rights. The continued treatment of their kids as a blank slate to impress their values on by any means necessary, including and especially abuse, is extremely, extremely important to conservatives.
Consent and respect for individual rights aren't high on the list or priorities to such people. It's "traditional" in the same way slavery was traditional.
Some would, some wouldn't. Personally I wouldn't have a problem with it. Better to have sex under my roof than somewhere less safe. In any case it's not like you need to worry about her getting pregnant.
That is all irrelevant to my point in any case. I would say that children's right to privacy with their counselor supercedes the parent's right to know what is said in their sessions.