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Two techniques to minimize the influence of parasitic losses on surface resistance measurements of superconductors employing dielectric resonators have been described. The first method optimizes the sapphire resonator aspect ratio to minimize the parasitic losses for given superconductor sample dimensions and measurement frequency. The second utilizes a reference resonator with a perfect conductor plane to cancel out the influence of parasitic losses by measurements of the resonant frequencies and Q-factors of two resonators. The second technique is recommended when materials having noticeable dielectric losses are used to construct dielectric resonators.