Fifty-five inclusive single nucleon-removal cross sections from medium mass neutron-rich nuclei impinging on a hydrogen target at ∼250 MeV/nucleon are measured at the RIKEN Radioactive Isotope Beam Factory. Systematically higher cross sections are found for proton removal from nuclei with an even number of protons as compared to odd-proton number projectiles for a given neutron separation energy. Neutron removal cross sections display no even-odd splitting, contrary to nuclear cascade model predictions. Both effects are understood through simple considerations of neutron separation energies and bound state level densities originating in pairing correlations in the daughter nuclei. These conclusions are supported by comparison with semimicroscopic model predictions, highlighting the enhanced role of low-lying level densities in nucleon-removal cross sections from loosely bound nuclei.