After some research, I have finally figured out where the Middletown Middies played their home games, Watt's Memorial Park. The Middletown Middies professional baseball team was part of the New York/New Jersey League in 1913 and the Atlantic League in 1914. In 1913, the league consisted of the Newburgh Dutchmen, Kingston Colonials, Danbury Hatters, Long Branch Cubans, and the Poughkeepsie Honey Bugs. In 1914, the Atlantic League was comprised of the Danbury Hatters, Bloomfield/Long Branch Cubans, Poughkeepsie Honey Bugs, Newark/Long Branch Cubans, Newburgh Hillclimbers, Paterson Silk Cities, and the Perth Amboy Pacers.
Special thanks to the Middletown Thrall Library for help with this information.