obviously i cannot speak for any of the dive shops. but the above statement may not always be the case. i would suggest discussing this with the instructor before you make plans just so you are not surprised and disappointed.
for myself, i rarely allow anyone on site if i have students. but it depends on the course we are doing, the student, and the other divers wanting to tag along.
i am sure whoever you decide to go with you all will have a great time. HAVE FUN !!
I am 100% sure BA allows a parent to be the "3rd wheel" so to speak for OW/AOW. It would be as an observer only. I did this with my son when he did his AOW with BA. Of course, Tech and other "specialty dives" would be the exception.
I asked Jeremy at Living Underwater about this as well, and he had no problem with it. You would need to book a dive of course to use each respective Op's tanks/weights. (Free shore diving if you are staying at BA and diving with them.)