The flight instructor reported that she briefed the soft-field takeoff procedure with her student while holding short for the active runway. After the flight was cleared for takeoff, the student pilot aligned the airplane with the runway center line, advanced the throttle to the full power position, and released aft pressure from the yoke. During the ground roll, the instructor briefly lost sight of the runway and when the runway reappeared, she observed the airplane rapidly veering to the left. The instructor applied full right rudder and right aileron control as they had lost directional control, but the airplane continued to the left. Unable to remediate the turn, the instructor reduced engine power, but quickly restored it with the intent of flying the airplane back to the runway as the airplane had just become airborne. However, the airplane rolled to the left, the stall horn came on and then they touched down on the taxiway and impacted a parked airplane. The right wing and left aileron were substantially damaged. The instructor and student reported no mechanical anomalies with the airplane that could have precluded normal operation.