In　this　paper,　a　one-step　protocol　is　proposed　for　the　parity　measurement　of　N　cat-state　qubits　which　are　encoded　on　the　cat　states　of　the　modes　in　superconducting　Kerr-nonlinear　cavities.　The　parity　measurement　is　performed　with　the　help　of　an　auxiliary　qutrit.　Especially,　the　auxiliary　qutrit　can　(cannot)　be　excited　to　the　higher-energy　levels　when　the　cat-state　qubits　are　in　the　even-　(odd-)　parity　state.　By　designing　the　Rabi　frequency　of　the　classical　fields　via　reverse　engineering　and　optimal　control,　the　qutrit　is　driven　to　an　excited　dressed　state　in　the　even-parity　case,　which　is　robust　to　the　systematic　errors　of　the　qutrit-cavity　coupling　strengths.　Accordingly,　the　parity　of　the　cat-state　qubits　can　be　distinguished　with　high　accuracy　by　measuring　the　final　population　of　the　ground　state　of　the　auxiliary　qutrit.　Numerical　simulations　also　show　that　the　protocol　is　insensitive　to　the　systematic　errors　of　the　classical　fields,　the　inhomogeneity　of　the　coupling　strengths,　intercavity　cross　talk,　unwanted　qutrit　transitions,　and　decoherence.　Therefore,　the　protocol　may　provide　an　effective　approach　for　parity　measurement　of　multiple　cat-state　qubits.　©　2024　American　Physical　Society.