In this paper, we study the numerical solution of singularly perturbed parabolic convection-diffusion type with boundary layers at the right side. To solve this problem, the backward-Euler with Richardson extrapolation method is applied on the time direction and the fitted operator finite difference method on the spatial direction is used, on the uniform grids. The stability and consistency of the method were established very well to guarantee the convergence of the method. Numerical experimentation is carried out on model examples, and the results are presented both in tables and graphs. Further, the present method gives a more accurate solution than some existing methods reported in the literature.