An efficient correction procedure via reconstruction for simulation of viscous flow on moving and deforming domains. In this paper, we report the development of a new parallel solver using the Correction Procedure via Reconstruction (CPR) for viscous flows on moving and deforming grids. By employing an accurate treatment of flux derivatives for moving and deforming unstructured grids consisting of all quadrilateral cells, it is found that the Geometric Conservation Law is not explicitly required, the free-stream preservation is automatically satisfied. The CPR code is verified using a benchmark case for a moving inviscid vortex on moving and deforming grids. The optimal orders of accuracy are obtained. It is subsequently employed to study viscous flows on moving and deforming grids. The CPR method is faster than and nearly as accurate as the SD method for solving viscous flow problems with moving boundaries.