The Milky Way's central parsec is a highly extinguished region with a population of high-proper-motion stars. We have tracked 145 stars for ~10yr at wavelengths between 1 and 4um to analyze extinction effects in color-magnitude space. Approximately 30% of this sample dims and reddens over the course of years, likely from the motion of sources relative to an inhomogeneous screen of dust. We correct previous measurements of the intrinsic variability fraction for differential extinction effects, resulting in a reduced stellar variability fraction of 34%. The extinction variability subsample shows that the extinguishing material has subarcsecond scales, much smaller variations than previously reported. The observed extinction events imply a typical cross section of 500 au and a density of around 3x10^4^atoms/cm^3^ for the extinguishing material, which are consistent with measurements of filamentary dust and gas at the Galactic center. Furthermore, given that the stars showing extinction variability tend to be more highly reddened than the rest of the sample, the extinction changes are likely due to material localized to the Galactic center region. We estimate the relative extinction between 1 and 4um as A_H_:A_K'_:A_L'_=1.67+/-0.05:1:0.69+/-0.03. Our measurement of extinction at longer wavelengths --L'(3.8um)-- is inconsistent with recent estimations of the integrated extinction toward the central parsec. One interpretation of this difference is that the dust variations this experiment are sensitive to --which are local to the Galactic center-- are dominated by grains of larger radius than the foreground.