Application of partial current-based coarse-mesh finite difference (pCMFD) acceleration to a one-node scheme is devised for stability enhancement of the parallel neutron transport calculation algorithm. Conventional one-node coarse-mesh finite difference (CMFD) allows parallel algorithms to be more tractable than two-node CMFD, but it has an inherent stability issue for some problems. In order to overcome this issue, pCMFD is modified to be fitted into the one-node scheme and is tested for both sequential and parallel calculations. The superior stability of the one-node pCMFD is shown by comparing results from analytic and numerical approaches. To investigate the convergence behavior of the acceleration methods in an analytic way, Fourier analysis is applied to an infinite homogeneous slab reactor configuration with the monoenergetic neutron flux assumption, and the spectral radius is calculated as a convergence factor. This paper carefully describes the process of the Fourier analysis on the parallel algorithm for neutron transport and compares it to that of the conventional sequential algorithm.