Paros and Antiparos are full of beaches, but it is best to ask the locals where there are beaches that aren't very touristic. Avoid those that are a quick walking distance from the town if you like a bit more privacy. There are many secluded beaches around Kolymbithres (also Kolimbithres). There are beaches that cover the area from Krios to St. Fokas to choose from.
In Naoussa and next to Damouli lies Ambelas, a quiet beach with very clean water and fresh fish. Between Santa Maria (a very popular beach) and Naoussa there are some quieter beaches.
Near Marpissa Pirgaki beach is very secluded, while Lolantonis and Glifa between Drios and Agaira are picturesque and worth visiting.