A class of singularly perturbed two-parameter parabolic partial differential equations with a large shift is studied. These equations include time-dependent variables and exhibit sensitivity to small perturbations in the system parameters. Moreover, the shift captures the influence of neighbouring states or events on the current evolution of the system. It adds memory-like behaviour to the problem, making their analysis and numerical solution more intricate. The solution of these equations exhibits a multiscale character. The interior and boundary layers exist across which the solution has a steep gradient. A higher-order accurate numerical method is proposed on a non-uniform mesh to solve the problem. The method combines an upwind difference scheme in space and a Crank-Nicolson scheme in time. The method is unconditionally stable, and parameters uniformly convergent with second-order accuracy in time and first-order accuracy in the space variable. Besides, the paper presents numerical results and illustrations to support theoretical estimates.