A novel parallel dynamic moving mesh algorithm designed for multiblock parallel unsteady flow calculations using body-fitted grids is presented. The moving grid algorithm within each block uses a method of arc-length-based transfinite interpolation, which is performed independently on local processors where the blocks reside. A spring network approach is used to determine the motion of the corner points of the blocks, which may be connected in an unstructured fashion in a general multiblock method. A smoothing operator is applied to the points of the block face boundaries and edges to maintain grid smoothness and grid angles. A multiblock parallel Euler/Navier-Stokes solver using multigrid and dual-time stepping is developed along with the moving mesh method. Computational results are presented for the unsteady flow calculations of airfoils and wings with deforming shapes as found in flutter simulations