Antiparos can be easily reached from Rafina (near Athens) or from Piraeus, where there are daily ferries going to the Cycladic islands. The journey to Paros takes about four to five hours, depending on the speed of the boat you take and the schedule it follows. After that you hop on the small ferry to Antiparos (30 minutes), or take a taxi to Pounda in the south of Paros, and take the car ferry there (10 minutes).
In 2009 the two small boats doing the Parikia - Antiparos Town trip have been sold. I was told that somebody else would take over somewhere in june so there would be a direct connection from the capital of Paros to Antiparos Town. In the meantime there are regular busses to the ferry in Pounta that will get you there in time, both when you go to Antiparos or when you go back.
On the airport of Athens there are regulair busses going to Rafina, and you don't have to drive through Athens itself. Off course you can also take a taxi. The ride takes about half an hour. It's also possible to fly to the island of Mykonos. From there it's only one and a half hours by ferry.