It sucks that the girl is progressing and enjoying herself more when mom's not around. I've never had any patience for those types who think they know more than the instructor/trainer when they've never been around horses a day in their lives. I don't mince my words ... that's why I don't give riding lessons, I tend to speak before thinking often, especially when my horses are concerned ... but I'd flat out tell this woman that she's getting in the way. Her daughter is progressing just fine without mommy dearest interfering and if mommy dearest can't shut up and listen to the person in charge (which, as it happens, is not her) then mommy dearest can find another instructor. Point. Blank. They're not paying you, so if they don't come it's not like you're going to lose any money, and on top of that this girl could potentially injure herself trying to show off, or injure your horse. I agree that horseback riding is great for kids, but it is a privilege not a right, and if the mother AND the daughter are not going to listen to you then they can find someone else's horse to ruin. I don't tolerate anyone treating my animals improperly - not my parrot, not my horse, not even my snakes.
But that's just me lol