This paper is concerned with the backstepping design of state feedback regulators that achieve robust output regulation for coupled linear parabolic partial integro-differential equations (PIDEs) with spatially varying coefficients. This problem is solved for a general setup, where polynomial and trigonometric reference inputs and disturbances are taken into account by employing a nondiagonalizable signal model. The regulator design is based on the internal model principle, which amounts to stabilize an ODE-PDE cascade, which consists of a finite-dimensional internal model driven by coupled parabolic PIDEs. For this, a systematic backstepping approach is developed and it is shown that the stabilizability depends on the plant transfer behavior. A simple proof of robust output regulation is given, which does not rely on solving the extended regulator equations. The results of the paper are illustrated by means of an unstable parabolic system described by three coupled parabolic PIDEs with two outputs. The robustness of the proposed state feedback regulator is verified by comparing it with a nonrobust feedforward regulator.