An investigation is made of the possibility of determining the intrinsic parities relative to the nucleon of K-Λ and K-Σ pairs by measuring the cross sections for photoproduction of these pairs from nucleons near threshold. The low-energy S and P wave contributions are calculated in a perturbation theory similar to that of Kawaguchi and Moravcsik. The question concerning the combination of nucleon and hyperon magnetic moments that is effective in the process is clarified. The calculated cross sections are quite different from those given in the long photon wavelength approximation by Feld and Costa. For most reasonable choices of the hyperon anomalous magnetic moments, certain calculated ratios of the close-to-threshold cross sections corresponding to different K-Λ and K-Σ charge states are quite sensitive to the K-hyperon parity. A model, in which the anomalous moments are the result of globally symmetric pion interactions, is used to illustrate the conclusions.