In this paper, a fitted Numerov method is constructed for a class of singularly perturbed one-dimensional parabolic partial differential equations with a small negative shift in the temporal variable.  Similar boundary value problems are associated with a furnace used to process a metal sheet in control theory. Here, the study focuses on the effect of shift on the boundary layer behavior of the solution via finite difference approach.  When the shift parameter is smaller than the perturbation parameter, the shifted term is expanded in Taylor series and an exponentially fitted tridiagonal finite difference scheme is developed. The proposed finite difference scheme is unconditionally stable.  When the shift parameter is larger than the perturbation parameter, a special type of mesh is used for the temporal variable so that the shift lies on the nodal points and an exponentially fitted scheme is developed.  This scheme is also unconditionally stable.  The applicability of the proposed methods is demonstrated by means of two examples.