An initial boundary value problem of convection-diffusion type for a singularly perturbed quasilinear parabolic equation is considered on an interval. For this problem we construct e-uniformly convergent difference schemes (nonlinear iteration-free schemes and their iterative variants) based on the domain decomposition method, which allow us to implement sequential and parallel computations on decomposition subdomains. Such schemes are obtained by domain decomposition applied to an e-uniformly convergent nonlinear base scheme, which is a classic difference approximation of the differential problem on piecewise uniform meshes condensing in a boundary layer. The decomposition schemes constructed in this paper converge e-uniformly at the rate of O(N ^-1 ln N + N₀^-1), where N and N₀ denote respectively the number of mesh intervals in the space and time discretizations.