In this article, the backstepping design of stabilizing state feedback controllers for coupled linear parabolic PDEs with spatially varying distinct diffusion coefficients as well as space and time dependent reaction is presented. The selected target system is a cascade of exponentially stable, time-invariant systems, with time dependent couplings, that is uniformly exponentially stable with a prescribed rate of convergence. To determine the state feedback controller, the kernel equations are derived, which results in a set of coupled PDEs for a time dependent and spatially varying kernel. For this, the method of successive approximations is extended from the time-invariant case to the present problem. The applicability of the method is demonstrated by the stabilization of two coupled unstable parabolic PDEs with space and time dependent coefficients.