This paper explores the $hp$-discontinuous Galerkin (DG) time-stepping method for nonlinear parabolic delay problems. Initially, we study $hp$-discontinuous Galerkin semi-discretization in time. We then study fully discrete schemes by applying continuous Galerkin and discontinuous Galerkin methods in space with $hp$-DG time stepping and get the existence of the discrete solutions using the fixed point theorems. We derive the optimal convergence for semi-discrete and fully discrete schemes $hp$-DG-CG ($hp$-DG in time and CG in space) and $hp$-DG-DG ($hp$-DG in time and DG in space) under various hypotheses on the regularity of the solution. A series of computational results are presented to demonstrate the effectiveness of the proposed schemes for constant and variable delays. We employ this scheme to illustrate the phase portrait of the Mackey-Glass equation and investigate cyclic competition dynamics in a reaction-diffusion model, highlighting the influence of delays.