In this paper, we study second-order necessary and sufficient optimality conditions of Karush-Kuhn-Tucker type for local optimal solutions in the sense of Pareto to a class of multiobjective optimal control problems with mixed pointwise constraints. To deal with the problems, we first derive second-order optimality conditions for abstract multiobjective optimal control problems which satisfy the Robinson constraint qualification. We then apply the obtained results to our concrete problems. The proofs of obtained results are direct and self-contained without using scalarization techniques.