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, due to their efficiency in 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. An important practical application of these results arises for image reconstruction in Parallel Magnetic Resonance Imaging (PMRI). Comparisons with existing optimization methods confirm the good performance of our approach for PMRI reconstruction.