Bears repeating.... put that on a bumper sticker; and I'll buy 2.
No, your child is not spying, anymore than another student who tapes their professor's words. That's my opinion.
If the prof is teaching the material the students have paid for, they most likely would not resort to taping what occurs in that classroom.
Professors who insist on using the classroom to infiltrate students' minds with their own philosophies about politics, morality, etc. need to be exposed. They are not doing the job that they are paid to do. I say, "Fire them". These profs have to wise up. So do parents footing the bills.