Prior to departure, the pilot visually inspected the fuel system of the powered-parachute and verified that the fuel tank was full. He departed from a pasture, flew to a nearby airport to practice a touch-and-go, and then was returning to their departure point. At approximately 1,000 feet above the ground (agl) the pilot noticed he was unable to hold his altitude and maintain airspeed. While in a shallow descent the pilot turned on the electric fuel pump and cycled the throttle. The engine responded to the inputs; however, would not achieve full power. Approximately 700 feet agl the pilot shut-down the engine over a field in which he was preparing to make an emergency landing. He began to perform a shallow S-turn maneuver to decrease his altitude. As he turned to the right the passenger compartment maneuvered outside the veil of the canopy above causing a rapid decrease in altitude; not able to make his selected landing site, the pilot elected to land straight ahead into the trees. The powered-parachute impacted several trees and substantially damaged the metal structure. Review of the carburetor icing probability chart revealed that the temperature and dew point were conducive to carburetor icing at cruise and glide power, the powered-parachute was not equipped with carburetor heat.