This paper considers the backstepping-based design of state feedback controllers for coupled linear diffusion-reaction systems with distinct constant diffusion coefficients, spatially-varying reaction and Neumann boundary conditions. A target system in cascade form with a local coupling term is proposed, which allows a direct specification of the stability margin of the resulting closed-loop system. By extending the method of successive approximations for the scalar case to the considered system class, the well-posedness of the resulting kernel equations is verified. This yields a systematic method for the backstepping control of the coupled parabolic systems in question, which is demonstrated for the stabilization of an unstable coupled system of two parabolic PDEs.