This article studies the numerical solution of singularly perturbed delay parabolic convection-diffusion initial-boundary-value problems. Since the solution of these problems exhibit regular boundary layers in the spatial variable, we use the piecewise-uniform Shishkin mesh for the discretization of the domain in the spatial direction, and uniform mesh in the temporal direction. The time derivative is discretized by the implicit-Euler scheme and the spatial derivatives are discretized by the hybrid scheme. For the proposed scheme, the stability analysis is carried out, and parameter-uniform error estimates are derived. Numerical examples are presented to show the accuracy and efficiency of the proposed scheme. © 2015 Elsevier Inc.