Confirming our previous proposal, we show within a continuum random-phase approximation calculation that the ratio of proton-to-neutron neutrino-induced quasielastic yield is a sensitive way to determine the strange quark axial form factor of the nucleon. Particularly, we find that the ratio is virtually independent of final state interactions (accounted for by a finite-range force derived from the Bonn meson-exchange potential) and nuclear structure effects. We present the ratio calculated for the neutrino beam available in an experiment currently approved at LAMPF.