The oldest monument in a New York park is the statue of George Washington in Union Square, erected in 1856. The subject and the form (an equestrian statue) are not very surprising. The second oldest, erected the following year, is more satisfyingly obscure: the General William Jenkins Worth Monument across the street form Madison Square. Worth is not very well known today, as his military career focussed on the War of 1812 and the Mexican-American War, both of which are shortchanged in popular history by the Revolution before and the Civil War after, but he was important enough in 1857 to get an obelisk in a public square as his grave marker. His most public legacy is the name of Fort Worth, Texas, after his former frontier post.
The engraving above is not an exaggeration: the dedication of the memorial had a crowd of thousands. Most public memorials to soldiers in parks are cenotaphs, but his is a grave marker. This made the 1914 construction of the portion of the BMT Broadway subway that passes immediately adjacent a bit more difficult than it would otherwise have been. I feel confident in saying that even if he were more famous, the subway would still have been built immediately adjacent to his resting place.