From Athens airport to the harbor:
From the airport of Athens to the harbor of Athens, Piraeus:
Departures just in front of the airport entrance every 20 minutes; approximately 1 hour to Piraeus.
Departures from the train station, just opposite of the airport; approximately 1 hour to Piraeus.
At Piraeus the bus stops near the train station, where just opposite (pass over the walking bridge) the boats to Paros will departure.
From Athens harbor, Piraeus to Paros:
In Piraeus you can take the ferry to Paros: departures twice a day; early in the morning or in the evening (17.30 hrs). It will take about 4 and a half hours to reach Paros with the normal ferry boat. There is also a High Speed boat .
Especially at the high season you better buy a ticket on-line from before (you can pick up the ticket just next to the boat).
A good alternative is to book a boat from Paros to Rafina instead of Piraeus (and also the other way around is possible).
Rafina is a small town half an hour by bus from the airport!
There is a boat which passes the Islands Mykonos and Tinos and arrives at the port of Rafina. From there, after 100 meters, there is the bus directly to the airport.
From Athens airport by plane