Hire a lawyer to write a lawyerly, very threatening letter stating that she will be taken to court for past board. Your parents don’t have to have a good case to actually win in court–there is no harm in bluffing. Deliver that via certified mail that she has to sign for. Don’t answer the phone when she calls, just let her stew on this. Then a few days later, drive to where she lives with the $1 bill of sale, all ready for her to sign. Tell her she’s going to lose in court, and when that happens, she probably will never be able to get a car loan or home mortgage.
Don’t leave until she signs.
If your parents can’t afford to get his teeth done, they probably can’t pay a lawyer. As a good daughter, you should offer to fund it. You can put up electric fence at home to divide the turnout and keep them separate, and bring your horse home for six months to get the money for this.
If you guys aren’t willing to do any of this, or it doesn’t work, I’d honestly question what would the real-world consequences be if they euthanized the horse and just lie and say it was colic. Put it this way-- if he really did colic and was in great pain, they actually would euthanize, wouldn’t they? And is your sister really going to take them to court over it? (This is very bad legal advice. Just saying that I would consider that solution, despite it being bad legal advice.)