In Wading River, the Wading River beach is open to residents as public beach in the town of Riverhead with proof of residency you can obtain a parking sticker and park at the beach. There is a bathing area in front of the life guard stand with an area that is roped off and for bathing. However, if you park in the lot with the permit and then walk on the access road immediately to the east of the parking lot you can walk your dog down to the water and then keep walking it east along the beach as long as you don't go any farther up towards the dunes than the last high water line. This is New York State Law that residences(and their leashed pets) have access to the beach as long as you stay between the water and the high tide water line (which you can see with line line of debris along the beach such as rocks, seaweed and driftwood, etc.)
Also, a number of Suffolk County Parks, including Smith Point Park in Shirley allow pets in certain areas, especially if you are camping there.