In Texas, the sheriff’s department normally handles cruelty to livestock. I would call them as I imagine California has cruelty laws pertaining to horses as well as others.
This may be a lunge whip — if so, it’s perfectly fine. A lunge whip is not used to hit the horse, but to direct it as the horse is lunged or free lunged (without a lunge line). if, however, the animals are being struck by a real whip, that is abuse.