Is it possible to start the engines while on push back?
Aircraft engines are started using the APU or a Airstart Cart. The GPU is for electrical power only and the engines are started off air power. It may be a Airstart was not available or they were running late on the block time and they blocked  using the APU to start the engines to make the block time. After engine start and before takeoff the crew will shut down the APU. Most US airlines start their engine off APU air unless the APU is not working then they will start off a Airstart Cart.