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We consider the $hp$-version discontinuous Galerkin finite element method ($hp$-DGFEM) with interior penalty for semilinear parabolic equations with locally Lipschitz continuous nonlinearity, subject to mixed nonhomogeneous Dirichlet–nonhomogeneous Neumann boundary conditions. Our main concern is the error analysis of the (spatially) semidiscrete $hp$-DGFEM on shape-regular spatial meshes. We derive error bounds under various hypotheses on the regularity of the solution, for both the symmetric and nonsymmetric versions of DGFEM.