After an engine run-up with the parking brake set, the first attempt to takeoff was aborted when the plane failed to accelerate satisfactorily. The brakes operated satisfactorily during the abort and a second run-up revealed no unusual engine indications. While taxiing back to the approach end of the runway, the pilot noted a lack of right braking action. He asked the passenger in the right front seat to check the right set of brakes, then he determined the brakes were acceptable. During a second attempt to takeoff, the pilot felt there was unsatisfactory acceleration and aborted again. However, he reported no braking action was available. He used aerodynamic braking and fishtailed to slow the aircraft, and finally, he ran into a snowbank at the departure end of the runway. An examination of the brakes only revealed evidence of excessive heat. Local maintenance personnel reported that the parking brake release trigger must be held in the depressed position while moving the brake handle forward to prevent inadvertent braking.