A lunge whip is like a cross between a bull whip and a buggy whip.
It is a proper tool for exercising or training a horse that does not have a rider. The horse is in a long line (lunge line) and is kept moving in a circle by following it with the whip. The whip is generally flopped along the ground behind the horse, occasionally making a sound. It is rarely, if ever, used to actually strike the horse, and never in an overhand motion.
Whatever they were doing was not proper...but as others have said may not constitute abuse. (and there story is likely to be a lot better than what you saw.)
Most animal abuse laws have exceptions for training stock animals. No blood, no foul is usually the way it will work.
I’m fond of my mare, and she thinks I’m the best human going, but I’ve made contact with her working her in a round pen. And there are horses I wouldn’t hesitate to make contact with, depending on their behavior.