He doesn't want to ride on his father's coattails, he wants to rely on his own effort.
Tips
culture
Internet buzzword that exploded around 2010 amid public anger over the 'My dad is Li Gang!' (我爸是李刚) incident, when a drunk driver invoked his official father's name to dodge consequences. Now standard slang for inherited privilege, the Chinese equivalent of 'born on third base'.
usage
拼 (pīn) here means 'to compete using' (as in 拼命 'risk one's life', 拼实力 'compete on strength'). So 拼爹 literally = 'compete with [one's] dad [as the asset]'.
