Could I use a powered parachute to fly around New York City?
Pretty simple, powered parachutes fall under the ultralight vehicle category, which is 14 CFR 103:

14 CFR 103.15 Operations over congested areas.:


  No person may operate an ultralight vehicle over any congested area of a city, town, or settlement, or over any open air assembly of persons.


And Also:

14 CFR 103.17 Operations in certain airspace.


  No person may operate an ultralight vehicle within Class A, Class B, Class C, or Class D airspace or within the lateral boundaries of the surface area of Class E airspace designated for an airport unless that person has prior authorization from the ATC facility having jurisdiction over that airspace.


It is highly unlikely that you would get a waiver to fly your PPC in the Manhattan Class-B, even if 103.15 didn't preclude it in the first place. You can fly in Class-B in certain areas, but you do need to be talking to ATC.