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In this paper we investigate vector-valued parabolic initial boundary value problems (A(t, x, D), B j (t, x, D)) subject to general boundary conditions in domains G in R n with compact C 2m -boundary. The top-order coefficients of A are assumed to be continuous. We characterize optimal L p -L q -regularity for the solution of such problems in terms of the data. We also prove that the normal ellipticity condition on A and the Lopatinskii-Shapiro condition on (A, B 1 , . . . , B m ) are necessary for these L p -L q -estimates. As a byproduct of the techniques being introduced we obtain new trace and extension results for Sobolev spaces of mixed order and a characterization of Triebel-Lizorkin spaces by boundary data.