Using offset-corrected Gaia-EDR3 parallax measurements and spectrophotometric methods, we have determined distances for 69 massive stars in the Carina OB1 association and associated clusters: Trumpler 16 (21 stars), Trumpler 14 (20 stars), Trumpler 15 (3 stars), Bochum 11 (5 stars), and South Pillars region (20 stars). Past distance estimates to the Carina Nebula range from 2.2 to 3.6 kpc, with uncertainties arising from photometry and anomalous dust extinction. The EDR3 parallax solutions show considerable improvement over DR2, with typical errors $sigma_{varpi}/varpi approx$~3-5%. The O-type stars in the Great Carina Nebula lie at essentially the same distance ($2.35pm0.08$ kpc), quoting mean and rms variance. The clusters have distances of $2.32pm0.12$ kpc (Tr 16), $2.37pm0.15$ kpc (Tr 14), $2.36pm0.09$ kpc (Tr 15), and $2.33pm0.12$ kpc (Bochum 11) in good agreement with the $eta$ Car distance of around 2.3 kpc. O-star proper motions suggest internal (2D) velocity dispersions $sim4$ km/s for Tr 14 and Tr 16. Reliable distances allow estimates of cluster sizes, stellar dynamics, luminosities, and fluxes of photoionizing radiation incident on photodissociation regions in the region. We estimate that Tr 14 and Tr 16 have half-mass radii $r_h = 1.5-1.8$ pc, stellar crossing times $t_{rm cr} = r_h/v_m approx 0.7-0.8$ Myr, and two-body relaxation times $t_{rh} approx 40-80$ Myr. The underlying velocity dispersion for Tr 14, if a bound cluster, would be $v_m approx 2.1^{+0.7}_{-0.4}$ km/s for $N = 7600^{+5800}_{-2600}$ stars. With the higher dispersions of the O-stars, mass segregation might occur slowly, on times scales of 3-6~Myr.
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