If the BO is using a boarder's horse for "lessons" then she better get permission.
That is a total breech of trust between the horse's owner and the BO.
Cat: The rule is more like only people in lessons or working at the barn can ride horses other than their on, even with permission from the owner. I totally would understand where the BO was coming from if what you said was the case, except any drama caused by kids riding horses without permission from the horse owner was because the BO had given the kid permission without the knowledge of the owner. I've seen this occur twice in the year and a half I've been there, and in one of the occasions it caused drama. In the other the horse owner didn't say anything because she's a 13 y.o.