We present fits to the broadband photometric spectral energy distributions (SEDs) of 158 eclipsing binaries (EBs) in the Tycho-2 catalog. These EBs were selected because they have highly precise stellar radii, effective temperatures, and in many cases metallicities previously determined in the literature, and thus have bolometric luminosities that are typically good to $lesssim$ 10%. In most cases the available broadband photometry spans a wavelength range 0.4-10 $mu$m, and in many cases spans 0.15-22 $mu$m. The resulting SED fits, which have only extinction as a free parameter, provide a virtually model-independent measure of the bolometric flux at Earth. The SED fits are satisfactory for 156 of the EBs, for which we achieve typical precisions in the bolometric flux of $approx$ 3%. Combined with the accurately known bolometric luminosity, the result for each EB is a predicted parallax that is typically precise to $lesssim$ 5%. These predicted parallaxes---with typical uncertainties of 200 $mu$as---are 4-5 times more precise than those determined by Hipparcos for 99 of the EBs in our sample, with which we find excellent agreement. There is no evidence among this sample for significant systematics in the Hipparcos parallaxes of the sort that notoriously afflicted the Pleiades measurement. The EBs are distributed over the entire sky, span more than 10 mag in brightness, reach distances of more than 5 kpc, and in many cases our predicted parallaxes should also be more precise than those expected from the Gaia first data release. The EBs studied here can thus serve as empirical, independent benchmarks for these upcoming fundamental parallax measurements.
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