hmm as a mother of sons, I would feel a bit worried about letting a 9 year old go in alone to the men's changing room.
If your private health club does not have family changing rooms, then I think the mother of the boy had little choice but to bring him in to the women's changing rooms with her.
Mind you, there should have been a notice displayed about children changing in single sex changing rooms, so you were not unclear about it.
I think your issue is with the health club, not the parent.
If there was a family room but the mother chose to not use it, then she was being (a bit) thoughtless IMO, but not highly unreasonable - 9 is still pretty young.
I can see it might possibly be more embarassing if the child was from your school, but she was hardly to know that.
I remember going to exercise classs and taking my 8 year old son with me. No creche facilities for him ( he was to old for them) so he used to sit at the side of the room playing with his gameboy while the class went on. Then a woman complained about having a boy watching her while she was exercising. The teacher asked me to leave him at home. I could only do that as I had a partner at home who could look after him. What if I had been a single mum without much childcare support?
I think there is a gross lack of facilites for women who want to exercise or swim and need to/want to take older children along with them. Sorry a bit off on a tangent....