I don't have children. But my answer is that, potentially, I would, but it would depend on the situation.

Firstly, I think the level of privacy that a child can reasonably expect to have from his parents is age and context-dependent. A thirty-year-old who has left the home has a far greater legitimate expectation of privacy than a fifteen-year-old living at home, who in turn can legitimately expect far more than a 5-year-old. I don't think most people have any problem with, say, using a baby-monitor on a young child, even though this could be viewed as a gross invasion of privacy if done on someone older.

Secondly, it is better to be honest and open where possible, as otherwise when your actions are discovered (and they likely will be), it could be seen as a breach of trust. However, if your child is lying to you, then it could be appropriate. For example, suppose my teenage child kept receiving suspicious parcels through the mail, and gave implausible accounts for the contents. I would then try to sit the child down and say right, we're going to open this together, and see what's inside. But if that wasn't possible, then yes, I might secretly open one of the parcels, to ascertain whether the child is doing something illegal, dangerous, or otherwise inappropriate.