Complex-valued data are encountered in many application areas of signal and image processing. In the context of optimization of functions of real variables, subspace algorithms have recently attracted much interest, owing to their efficiency for solving large-size problems while simultaneously offering theoretical convergence guarantees. The goal of this paper is to show how some of these methods can be successfully extended to the complex case. More precisely, we investigate the properties of the proposed complex-valued Majorize-Minimize Memory Gradient (3MG) algorithm. Important practical applications of these results arise in inverse problems. Here, we focus on image reconstruction in Parallel Magnetic Resonance Imaging (PMRI). The linear operator involved in the observation model then includes a subsampling operator over the $k$-space (spatial Fourier domain) the choice of which is analyzed through our numerical results. In addition, sensitivity matrices associated with the multiple coil channels come into play. Comparisons with existing optimization methods confirm the good performance of the proposed algorithm.