In this paper, a multi-aperture multiplexing multiple-input multiple-output (MAM-MIMO) sparse array is presented for cooperative automotive radars (CARs). The proposed sparse array composed of multiple subarrays can simultaneously cover a wide field-of-view (FOV) and achieve the required azimuth resolution at different ranges. To validate this idea, an optimization model for the MAM-MIMO sparse array is derived based on the example of CARs. This optimization model has been found by combining the peak-to-sidelobe ratio (PSLR) at all beams pointing within the constraints of different detection ranges. In addition, a hierarchical genetic algorithm based on the multi-objective decomposition method has been developed to obtain the optimized sparse array. The proposed method has been evaluated through both simulations and experiments. It is demonstrated that the optimized MAM-MIMO sparse array can effectively suppress sidelobes of its subarrays, yet with reasonably high azimuth resolutions and large FOVs.