This study designs and investigates a robust numerical method for a coupled system of singly perturbed parabolic delay partial differential equations. Design a priori bounds are derived on the regular and layer components of the solution and their derivatives. Based on these a priori limitations, Shishkin and generalized Shishkin type appropriate layer adaptive meshes are created in the spatial direction. Then, the issue is discretized utilizing a layer-adapted Shishkin and generalized Shishkin type meshes in the spatial direction and an implicit Euler mesh in the temporal direction. It is suggested to tackle the problem numerically, taking into account both generalized Shishkin meshes and layer-appropriate Shishkin meshes.