Previous analyses of various standard candles observed by the Gaia satellite have reported statistically significant systematics in the parallaxes that have improved from ∼250 μas in the first data release (DR1) to 50-80 μas in the second data release (DR2). Here we examine the parallaxes newly reported in the Gaia early third data release (EDR3) using the same sample of benchmark eclipsing binaries (EBs) we used to assess the DR1 and DR2 parallaxes. We find a mean offset of -37 ± 20 μas (Gaia - EB), which decreases to -15 ± 18 μas after applying the corrections recommended by the Gaia Mission team; global systematics in the Gaia parallaxes have clearly improved and are no longer statistically significant for the EB sample, which spans 5 ≲ G ≲ 12 in brightness and 0.03-3 kpc in distance. We also find that the Renormalized Unit Weight Error (RUWE) goodness-of-fit statistic reported in Gaia DR3 is highly sensitive to unresolved companions (tertiaries in the case of our EB sample) as well as to photocenter motion of the binaries themselves. RUWE is nearly perfectly correlated (r2 = 0.82) with photocenter motions down to ≲0.1 mas, and surprisingly this correlation exists entirely within the nominal "good" RUWE range of 1.0-1.4. This suggests that RUWE values even slightly greater than 1.0 may signify unresolved binaries in Gaia, and that the RUWE value can serve as a quantitative predictor of the photocenter motion.