A small list of possible activities:

Kite Surfing
at Pounda beach (in front of Antiparos): http://www.paroskite-procenter.com/

Wind Surfing
at Santa Maria Beach (Naoussa)
at Golden Beach and New Golden Beach (east coast)

Scuba Diving
at Pounda (south of Parikia, the town opposite Antiparos): www.eurodivers.gr

Sea discovery programs
with snorkeling and diving for kids, teens, families and schools
(situated at Golden Beach, check link: www.octopuseatrips.com)

Water Ski, Tubes
at several different beaches like Kolimbithres during high season mostly

Horse Riding

Tours with Fishing Boats

Sailing
Daily tours & more

Sightseeing Tours

Daily trips to other Islands
like Naxos, Mykonos, Delos, Santorini, departing from Naoussa, Pisso Livadi

Beach Volley, Bunjee Jumping, Beach Bar, Swimming Pool
go to Pounda Beach

Mountain Biking Tours
organized (Naoussa) or "rent a bike"

Fitness Centers / Body Building
in Parikia & Naoussa