The pilot of a Piper L-4 airplane reported that shortly after he started the taxi roll from the parking area, he initiated a shallow turn toward the taxiway. As he reached the taxiway he steered to follow the centerline, however, the airplane continued to turn to the right. He reported, in part, that the left brake inputs were not responsive and the airplane continued to the right and collided with a standing occupied Cessna. Both airplanes sustained substantial damage, but neither the solo airline transport pilot of the Piper, nor the commercial pilot and five passengers (skydivers) aboard the Cessna, were injured. Post accident examination of the Piper's brake assembly by inspectors from the Federal Aviation Administration revealed no evidence of abnormal wear or fluid leaks. The inspectors performed a function test on the brake systems and reported the assemblies did not properly engage when the brake pedals were actuated. The inspectors reported that the brake fluid reservoirs were low. The right side reservoir contained approximately three-quarters of total capacity and the left side reservoir contained approximately one-half of total capacity.