Spark gap switch (SGS) is a fundamental but critical component for large-scale pulsed power devices, whose reliable operation is significantly affected by the breakdown characteristics of SGS. It is observed experimentally that, with the increase of overvoltage, the bridging position of the spark channel transits from edge to stochastic center. In this work, the influence of overvoltage on the breakdown process of a parallel-plate SGS with low geometric distortion of static electric field (< 13%) between an atmospheric-pressure air gap of 5 mm is investigated by particle-in-cell/Monte Carlo collision simulation. It is found that, under a low overvoltage (ratio of applied voltage Ua to static breakdown voltage U0, Ua/U0=1.5), the streamers at the edge first bridge the gap before those in the central region, due to the field enhancement induced by the electrode curvature. Under higher overvoltage (Ua/U0=3), the synchronicity between streamers initiating from the center and those from the edge is greatly improved during the inception stage. After the streamers pass the middle of the gap, the field enhancement at the streamer front is more intensified and promotes the generation of fast electrons. These fast electrons rapidly magnify the difference among the propagating streamers by providing abundant seed electrons ahead of the discharge channel, which leads to the randomness of the bridging position. The results in this work demonstrate the relationship between the overvoltage and the streamer dynamics, which is beneficial for the performance improvement of SGS.