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In this paper, utilizing techniques in compressed sensing, parallel optimization and deep learning, we propose a model-driven approach to jointly design the common measurement matrix and GROUP LASSO-based jointly sparse signal recovery method for complex sparse signals, based on the standard auto-encoder structure for real numbers. The encoder achieves noisy linear compression for jointly sparse signals, with a common measurement matrix. The GROUP LASSO-based decoder realizes jointly sparse signal recovery based on an iterative parallel-coordinate descent (PCD) algorithm which is proposed to solve GROUP LASSO in a parallel manner. In particular, the decoder consists of an approximation part which unfolds (several iterations of) the proposed iterative algorithm to obtain an approximate solution of GROUP LASSO and a correction part which reduces the difference between the approximate solution and the actual jointly sparse signals. The proposed model-driven approach achieves higher recovery accuracy with less computation time than the classic GROUP LASSO method, and the gain significantly increases in the presence of extra structures in sparse patterns. The common measurement matrix obtained by the proposed model-driven approach is also suitable for the classic GROUP LASSO method. We consider an application example, i.e., channel estimation in Multiple-Input Multiple-Output (MIMO)-based grant-free random access which is proposed to support massive machine-type communications (mMTC) for Internet of Things (IoT). By numerical results, we demonstrate the substantial gains of the proposed model-driven approach over GROUP LASSO and AMP when the number of jointly sparse signals is not very large.