They were rude, period. It sounds like neither the mother nor the student really consider lessons a priority.

Maybe tell them your schedule is now full, since you filled the spot that this no-show usually takes. You will put them on a waiting list if they want to re-book.

Really unprofessional and discourteous on their part, and a waste of your time and gas (heck, I would bill them for the fuel alone!).

I think the student HAS quit. You just need to make it official.

JMO - I would have phoned first, then emailed. A lot of folks don't live on their computers like I do.