On April 11, 1995, at 1404 central daylight time, one passenger was seriously injured and two passengers received minor injuries aboard a Boeing 757-223, N683A, during taxi at the San Antonio, Texas, International Airport. Six crew members and 123 passengers were not injured. Visual meteorological conditions prevailed for American Airlines Flight 1308 from Dallas/Fort Worth (DFW), Texas, which was conducted under Title 14 CFR Part 121. Crew and company personnel reported the information in this paragraph. After arriving at the airport, the captain taxied the airplane into position at the gate, pulled the parking brake handle in the full "UP" position and checked the parking brake light on the annunciator panel. The seat belt sign was turned "OFF" and the crew proceeded with the engine shutdown checklists. Upon hearing a "thud", the captain "looked up to see the aircraft moving" and stopped the airplane with the brakes. He was informed, by the flight attendant, of the three passenger injuries. Ground personnel threw chocks at the airplane as it rolled forward, trying to alert the flight crew. Maintenance personnel examined the airplane and "ascertained that no brake failure had occurred." The airplane was released for continued service with no discrepancies reported.