The minimal surface equation for timelike surfaces of the Minkowski space is a quasi-linear wave equation. The nonlinear part of the equation exhibits a cancellation known as the null condition. We replicate the strategy of Smith and Tataru of constructing a wave-packet parametrix, which coupled with a space-time estimates for the null form allows us to lower the regularity compared to the general result for the quasilinear wave equation obtained by Smith and Tataru.