Iterative self-consistent parallel imaging reconstruction (SPIRiT) was an autocalibrating model for parallel magnetic resonance imaging reconstruction, which is often formulated as a SPIRiT reconstruction problem with some regularization terms. Some methods based on the operator splitting and alternating direction method of multipliers (ADMM) have been employed to solve the formulated regularized SPIRiT problem. In this paper, we propose to combine the sparsifying transform learning and joint sparsity with Cartesian SPIRiT parallel magnetic resonance imaging, and solve the resulting reconstruction problem by using the variable splitting and ADMM techniques. Simulation experiments on four in vivo data sets demonstrate that the proposed algorithm achieves a better image reconstruction quality than that of other competing methods. In addition, the proposed algorithm is very suitable for graphics processing unit (GPU) parallel computing, and its accelerated version, achieved by using a GPU, is very fast, requiring only 6.7 s to reconstruct a 200 x 200 pixel image with 8 channels.