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Existing parallel MRI methods are limited by a fundamental trade-off in that suppressing noise introduces aliasing artifacts. Bayesian methods with an appropriately chosen image prior offer a promising alternative; however, previous methods with spatial priors assume that intensities vary smoothly over the entire image, resulting in blurred edges. Here we introduce an edge-preserving prior (EPP) that instead assumes that intensities are piecewise smooth, and propose a new approach to efficiently compute its Bayesian estimate. The estimation task is formulated as an optimization problem that requires a non-convex objective function to be minimized in a space with thousands of dimensions. As a result, traditional continuous minimization methods cannot be applied. This optimization task is closely related to some problems in the field of computer vision for which discrete optimization methods have been developed in the last few years. We adapt these algorithms, which are based on graph cuts, to address our optimization problem. The results of several parallel imaging experiments on brain and torso regions performed under challenging conditions with high acceleration factors are shown and compared with the results of conventional sensitivity encoding (SENSE) methods. An empirical analysis indicates that the proposed method visually improves overall quality compared to conventional methods.