This is complete and utter nonsense. And it undermines whatever credibility your other statements might have had.
Surely there is a continuum from “no interference with the animal” all the way to “constant and damaging physical abuse”. PETA purports to advocate the former. No one defends the later. There is a large but excluded middle.
Does abuse occur? Surely it does; there are nasty people in the world. Is it common practice in circuses, dog training, and/or horseback riding (just to use examples already noted)? Hardly. Why? Because it is counter productive. Animals cannot be punished into performing behaviors. Punishment only produces avoidance reactions. Abusive treatment cannot produce a performing animal, whether that be a circus elephant, a field trial retriever or an Olympic dressage competitor. Regardless of the propensities of the human perpetrator, abuse simply does not work as a training technique.
Might someone simply get mad and beat their horse, their dog, or even their elephant? Assuredly yes. Would such a person last long as a dog trainer, a horseback rider, or a mahout? Hardly. Animals remember frightening and painful experiences very well. Anticipating additional unpleasantness, they become shy, furtive, uncooperative, even dangerous.
Not that systematic abusive treatment of dogs (e.g., Vick) has exactly this result. The dogs are not “trained”. Rather they are simply deranged and pathetic passive - aggressive creatures that can only “perform” one job-- psychotic fighting.
It is hardly reasonable to identify an extreme instance, something from the fringe of the bell curve, and use it to paint all and sundry. Still, this is a common and often effective political tool. One that PETA frequently uses, to its advantage.