In Paroikia:

Elaea (the Olive), on the seafront, good Greek and seafood menu

In town, off of Market Street, both Levantis and Tamarisko are two of the more upscale restaurants in town, both have very good menus and nice outdoor terraces. A newer, also more upscale restaurant in town is Franca Scala, on a balcony in old part of off Market Street.

Albatross, on the seafront of Paroikia, variety of traditional Greek food & fish

 In Naoussa:

Barbarossa, on the seafront, at the north end, overlooking the small harbor - upscale and very well-known for fine, fresh seafood, lovely outdoor setting

Meltemi, at the south end of the seafront, with an owner from Crete, good classic Greek and Cretan food

 For beach/seaside eating:

Magaya Restaurant, right on great little beach bay, just a kilometer south from Paroikia town, serves, both Greek and some Asian food, good simple lunch after the beach and for swimming, also, good all day hangout place, couches, hammocks, etc

at Parasporos beach (south of Paroikia), two nice beach restaurants:

Parasporos Beach Taverna, owned by an English/Scottish couple, continental cuisine

Makis Taverna, on the beach, good seafood and classic Greek dishes (well-known for their sun-dried mackerel and fresh calamari)

 Aliki:

Captain Peppino's: Traditional beach side taverna, open all year round. meat plates, fresh lobster and crabs, tasty Greek-style appetizers and complimentary freshly made deserts, right on the sea.

 Pounta:

 Thea - at the small port of Pounta (where the ferry from Paros to Antiparos goes) -- one of the nicer restaurants on Paros with an experienced sommelier/owner with extensive wine cellar.

Antiparos:

 If you do go on a daytrip to neihboring Antiparos (only 5 min by ferry), Stillwaters, on the southeast seaside of Antiparos (facing Paros), is a very nice beach taverna/restaurant - and Tom Hank's favorite spot to eat out when he's at his villa just up the road on Antiparos.