Pulse compression can enhance both the performance in range resolution and sensitivity for weather radar. However, it will introduce the issue of high sidelobes if not delicately implemented. Motivated by this fact, we focus on the pulse compression design for weather radar in this paper. Specifically, we jointly design both the subpulse codes and extended mismatch filter based on the alternating direction method of multipliers (ADMM). This joint design will yield a pulse compression with low sidelobes, which equivalently implies a high signal-to-interference-plus-noise ratio (SINR) and a low estimation error on meteorological reflectivity. The experiment results demonstrate the efficacy of the proposed pulse compression strategy since its achieved meteorological reflectivity estimations are highly similar to the ground truth.