We revisit the Cauchy-Dirichlet problem for degenerate parabolic scalar conservation laws. We suggest a new notion of strong entropy solution. It gives a straightforward explicit characterization of the boundary values of the solution and of the flux, and leads to a concise and natural uniqueness proof, compared to the one of the fundamental work [J. Carrillo, Arch. Ration. Mech. Anal., 1999]. Moreover, general dissipative boundary conditions can be studied in the same framework. The definition makes sense under the specific weak trace-regularity assumption. Despite the lack of evidence that generic solutions are trace-regular (especially in space dimension larger than one), the strong entropy formulation may be useful for modeling and numerical purposes.