Thanks for your question.
Article 26, Section 377 of New York Agriculture and Markets states that dead livestock must be buried as follows:
§ 377. Disposal of dead animals. (1) The carcasses of large domestic animals, including but not limited to horses, cows, sheep, swine, goats and mules, which have died otherwise than by slaughter, shall be buried at least three feet below the surface of the ground or otherwise disposed of in a sanitary manner by the owner of such animals, whether the carcasses are located on the premises of such owner or elsewhere. Such disposal shall be completed within seventy-two hours after the
owner is directed to do so by any peace officer, acting pursuant to his special duties, police officer, or by a designated representative of the commissioner.
(2) Notwithstanding section forty-one of this chapter, any violation of this section shall constitute a violation. This section shall not apply to animal carcasses used for experimental or teaching purposes.
So long as you bury the horse at least three feet below the ground surface, you can legally bury a dead horse in your pasture in New York.
Hope this helps.