MRI is an imaging technology that non-invasively obtains high-quality medical images for diagnosis. However, MRI has the major disadvantage of long scan times which cause patient discomfort and image artifacts. As one of the methods for reducing the long scan time of MRI, the parallel MRI method for reconstructing a high-fidelity MR image from under-sampled multi-coil k-space data is widely used. In this study, we propose a method to reconstruct a high-fidelity MR image from under-sampled multi-coil k-space data using deep-learning. The proposed multi-domain Neumann network with sensitivity maps (MDNNSM) is based on the Neumann network and uses a forward model including coil sensitivity maps for parallel MRI reconstruction. The MDNNSM consists of three main structures: the CNN-based sensitivity reconstruction block estimates coil sensitivity maps from multi-coil under-sampled k-space data; the recursive MR image reconstruction block reconstructs the MR image; and the skip connection accumulates each output and produces the final result. Experiments using the fastMRI T1-weighted brain image dataset were conducted at acceleration factors of 2, 4, and 8. Qualitative and quantitative experimental results show that the proposed MDNNSM method reconstructs MR images more accurately than other methods, including the generalized autocalibrating partially parallel acquisitions (GRAPPA) method and the original Neumann network.