We constrain the parameters of neutron superfluidity in the cores of neutron stars making use of the recently proposed effect of resonance stabilization of r modes. To this end, we, for the first time, calculate the finite-temperature r-mode spectra for realistic models of rotating superfluid neutron stars, accounting for both muons and neutron-proton entrainment in their interiors. We find that the ordinary (normal) r mode exhibits avoided crossings with superfluid r modes at certain stellar temperatures and spin frequencies. Near the avoided crossings, the normal r mode dissipates strongly, which leads to substantial suppression of the r-mode instability there. The extreme sensitivity of the positions of avoided crossings to the neutron superfluidity model allows us to constrain the latter by confronting the calculated spectra with observations of rapidly rotating neutron stars in low-mass x-ray binaries.