I always mixed Parasailing and Paragliding till yesterday (25th September 2010), when I myself did parasailing at Lake Tahoe (California) USA.
I found the following answer:
Parasailing, also known as parascending, is a recreational activity where a person is towed behind a boat attached to a specially designed parachute, known as a parasail. The harness attaches the pilot to the parasail, which is connected to the boat, or land vehicle, by the tow rope.
Paragliding is a recreational and competitive flying sport. A paraglider is a free-flying, foot-launched aircraft. The pilot sits in a harness suspended below a fabric wing, whose shape is formed by its suspension lines and the pressure of air entering vents in the front of the wing. Paragliding can be powered by a small engine with a propeller.