There are two beaches in Parikia, although neither is very picturesque. The main one is up north of town. Follow the blacktop road to the beach. It is rocky and crowded, with little to view. But it is nice for the end of the day and little dip to cool down. Not much of surf either.
Khryssi Akti (Golden Beach) is a beautiful sandy beach on the other side of the island. It is south of the city of Naoussa and takes quite a while to get there. You will need to rent a scooter or a car to get there. The day we were there was quite windy, which blew quite a bit of sand in our faces. But the view is gorgeous and the water as clear as could be. They do rent lounge chairs with umbrellas there for five Euros. There are no immediate public restrooms although there is a bar/restaurant near the beach.