We describe a method to quantify the degree of parallelism between two transparent glass mirrors spaced a few microns apart. Our technique, which permits the measurement and correction of deviations as small as l/10000 from parallelism, is fundamental to the successful operation of tunable narrowband interference filters for two reasons. First, the highest throughput is achieved when the plates are parallel at any plate spacing. Second, the lowest resolution (largest bandpass) imaging is achieved when the plates are only a few microns apart, but there is a real danger of the plates touching if parallelism is not maintained. The Taurus Tunable Filter (TTF) 3 is a Fabry-Perot cavity with an adjustable plate spacing of 2-13 mm. The parallelism measurement involves repeated imaging through a focal-plane slit and a series of pupil-plane masks. This approach is particularly efficient when the plate scanning is synchronized with the movement of the charge on the CCD. We assess the effects of wavelength-dependent phase changes within the inner surface coatings of the plates. These become important as the plates approach a spacing comparable in size to the thickness of the coatings.