I have been on Paros in June nearly every year for the last 15 years or so, and have never had a problem getting around by bus. Obviously you can't see everything in a three night stay, but several Fodorites have given you some ideas from which to choose. If those three nights also mean three full days, you shouldn't have a problem, for example, visiting Naoussa one day, Lefkes and Piso Livadi a second day, and Antiparos a third. If you have only two full days, then something will have to give.

As Mariha wrote, the bus timetable on Paros changes about every two weeks. By coincidence I happen to have a copy of the KTEL Paros printed timetable from 18th June last year:

In the last two weeks of June on Route #2 were 16 daily buses between Parikia and Naoussa, the first leaving Parikia at 0740, and the last returning from Naoussa at 0000 (midnight).

Route #1, the Parikia to Drios route (going through Lefkes and Piso Livadi), had buses leaving Parikia from 0730, and returning to Parikia as late as 1900. I see no reason why you couldn't take a morning bus from Parikia to Lefkes, explore that small village, then get on a later bus to e.g. Piso Livadi. In the afternoon take a bus back to Parikia, insuring you get on one that leaves Drios at 1700 or 1900.

Likewise, in late June there were numerous buses on Route #4 going from Parikia to Pounta port, connecting with the ferry to Antiparos. The first left Parikia at 0710, and the last returned from Pounta to Parikia at 2325. You also have an option to take the excursion boat to Antiparos directly from Parikia harbour, but you will have to check the timetables when you get to Paros. :-)

Hi indiancouple, your head will be swimming with all these ideas, but I'll try to narrow the scope for you. On Paros, you can easily spend your a day wandering around Parakia, visiting the Church of 100 Doors, the museum, the Old Market area.
You could spend another full day touring Leftkes, Naoussa, and Alyki. Bus service is very good, but renting a car gives you more flexibilty. I have never had any issues renting cars there - or taking the bus for that matter.
Swimming will be lovely in late June. If you want to beach it, you may need to forget seeing all the other parts of the island. You can take a short boat trip from the port in Parakia to Krios beach across the harbor - fun. If you hire a car, drive to Naoussa then to Kolymbithres Beach - wonderful! There are also great scenic hiking trails at a nature area on the road beyond the beach.
Antiparos is also a wonderful day trip, but you would have to give up any other sightseeing on Paros. I recommend you save this for a longer visit as there is so much to do on Paros.

Thanks HappyTrvlr, mariha, Heimdall and Parosblue for your wonderful inputs. I was traveling, so could not respond earlier. Great info, and very very useful. Now I think we have enough perspectives on Paros, and no shortage of things to do. Thanks once again.

check the nearby activities of the following pages
http://www.hotelapollon.gr/facilities.php
http://www.hotelapollon.gr/parikia_location.php
http://www.hotelapollon.gr/paros.php
http://www.parosweb.com