While the most famous park accessible by transit is probably Glacier National Park on the Empire Builder, there are some closer to NYC that are accessible.
Notably Lake Champlain and a good portion of the Adirondacks are accessible by both the Adirondacks line and the Ethan Allen Express.
The Delaware Water Gap is harder to access but The Main and Bergen Line from NJTransit can take you from Hoboken to Suffern where you can transfer to the Metro North Port Jarvis Line. In Port Jarvis you can catch a bus to the Gap