I did all of my work alone with kids (though I don't work with kids now). Working with the parents in the room often changes the child's response to the therapist. Also when assessing for abuse, you aren't likely to get an honest answer from the child if the parent is in the room. Parents also often try to speak for the child or encourage them when they are resistant in session. They really need to just disappear into the background.

That said, I'm not vouching for the psychologist either. I can understand your concern, but it is quite normal for the therapist to work alone with the kid.

Most therapists have recording tech. Heck, laptops and phones can record, and it shouldn't be too much to ask for them to record their session for you. If they decline that, I would have a problem with it.