The pilot reported that during the takeoff roll from the short dirt airstrip, the airplane encountered a gust of wind. He attempted to apply corrective control inputs; however, the airplane departed the runway and struck a powered parachute, which was parked on the adjacent taxi area. The pilot elected to continue the takeoff, and flew an additional hour to his destination airport. Witnesses reported that soon after the airplane began its takeoff roll, it swerved onto the taxiway, continuing its takeoff attempt almost to the end of the runway, where it struck the powered parachute. The powered parachute was destroyed during the accident sequence, and the airplane sustained substantial damage, which included a 12-inch-wide gash in the fuselage structure stretching from the baggage door to the aft of the tailcone. Both witness reports, and official weather observations, indicated that strong quartering tailwind conditions existed on the departure runway at the time of the accident. The pilot reported no preimpact mechanical malfunctions or failures with the airframe or engine that would have precluded normal operation.