The dynamics of transient plasma breakdown phase in conventional spark plugs play an important role in determining electrode damage. Despite the importance of the breakdown event, there are few computational simulations of spark-plug arcs reported in the literature that focus on the non-equilibrium breakdown phase. The goal of this work is to investigate the preliminary plasma breakdown phase of the arc, during which the plasma dynamics are highly non-equilibrium in nature. We utilize VizGlow, a high-fidelity non-equilibrium plasma software tool, to simulate the plasma breakdown phase during the first several nanoseconds of the discharge event. During the early breakdown phase, the plasma forms in thin filamentary streamers which provides the initial conductive pathway across the gap. Once the streamer channels have bridged the gap, the plasma begins to transition to a thermal arc. The redistribution of electrostatic potential across the gap during the breakdown phase causes a large voltage drop near the cathode. This leads to significant ion bombardment on the electrode surface which is a prime source of electrode erosion.