In this paper, we present a novel parallel high-order sliding and deforming spectral difference (SD2) method for solving the two-dimensional compressible Navier-Stokes equations on unstructured quadrilateral grids. In the SD2 method, an arbitrary Lagrangian-Eulerian (ALE) approach is combined with a novel sliding interface approach to separately handle rotational and translational motions on a dynamic grid. The SD2 method is shown to be capable of completely removing grid skewness caused by rotating boundaries to improve solution qualities. The SD2 method is verified as high-order accurate and is scalable for parallel computing on a distributed-memory computer. The SD2 method is also proven to be more robust and more accurate than deforming-only ALE approach for problems involving large-amplitude rotating motions. (C) 2016 Elsevier Ltd. All rights reserved.