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We consider a non-monotone FEM discretisation of a singularly perturbed time-dependent reaction-diffusion problem whose solution exhibits strong parabolic layers. We conduct a stability and convergence analysis for arbitrary meshes. The method is shown to be maximum-norm stable although it is not inverse monotone. The convergence result implies convergence of the method on Shishkin and Bakhvalov meshes uniformly in the perturbation parameter e. Numerical experiments complement our theoretical results.