There’s no specific name for vegetarians who ride animals. Tipically, vegetarians don’t eat land-based meat like beef, pork, chicken, etc. However; vegetarians who ride animals do eat fish,–if supported. For this instance, you visited a farm where a vegetarian who rides a cow on a farm where dairy cattle are raised, but only few cattle. Or a vegetarian is riding a horse on a horse farm. There are some vegetarians who ride any animal as their mode of traveling.
I define vegetarians who ride animals as “mounted vegetarians” because, a vegetarian has an ability to ride animals. There are many people who ride animals around the world. Some people ride cattle, reindeer, water buffalos, ostriches, and other animals that are bred for riding. At the other hand; some people ride elephants, but many mahouts has been abusing elephants for years, after they’ve been taimed. Painful methods of training elephants has been linked to mahouts being killed by their own beasts who were been raised by owners. There’s no way for mahouts to train elephants with alternatives. Mounted vegetarians don’t rely on painful methods of training animals to be ridden. It takes time, and patience. Riding animals can be fun, but if you want to ride an animal, you must take time training her, and treat her with dignity. Whether if your animal is furry or not; you have to take your time.
If you were a mounted vegetarian who ride animals; you must avoid relying on painful methods of training animals for riding. You can try riding methods, such as dressage, dressage is an art of riding an animal.
In my opinion; there’s no technical name for a vegetarian who ride animals.