I’m wonder if the parents ditched the horse/made other arrangements because the barn put their foot down with the parents. I’m not so sure it will go as well for you as it did the barn. My experience is that this type of behavior usually gets worse before it gets better or goes away. In this case, they just directed the behavior onto someone else with a lot less recourse than a boarding barn has. They will care more about keeping their arrangement so they can ditch their kid off than they will about doing the right thing.
I would not be surprised if they threaten or try to sue you for “enforcement” or to take the horse back (so they have leverage to do this to someone else) as a means to pressure you into continuing the arrangement as is. Also, there are other ways to make your life hell in the furtherance of making you cave, be prepared to have to deal with them. I would consult with an attorney about how solid my claim was on the horse in your jurisdiction, what legal obligations you have about the visits, and what recourse you have in the case they continue to try to force the issue by dropping the kid off unannounced, and advice on how to word any such cease and desist communication you send them. If a conversation/letter from you goes badly or is ignored, be prepared to have a lawyer write them a letter to back you up. Let the attorney warn them of any possible legal outcomes with respect to their behavior of just dropping their kid off on you without your expressed permission.
These people wouldn’t acknowledge to their kid that they sold the horse. That is a moral low that you should recognize. Grifters are good at targeting people who will put up with their crap. Don’t take it from them anymore. I hope they will mosey on off relatively easily, but be prepared to act if you have a psychopath declare war on you in an effort to force you to cave to their whims.