  CONTEXT. Gaia EDR3 has produced parallaxes for 1.468x10^9 sources but there
are calibration issues that require corrections to the published values and
uncertainties. AIMS. We want to characterize the behavior of the uncertainties
of the Gaia EDR3 parallaxes. We also aim to provide a procedure for the
calculation of distances to stars and stellar clusters. METHODS. We reanalyze
some of the data in the calibration papers for QSO and LMC parallaxes and
combine those results with measurements for six bright GCs. We calculate the
angular covariance of EDR3 parallaxes at small separations based on the LMC
results and combine it with the results for larger angles using QSOs to obtain
an analytical formula for the angular covariance over the whole sky. The
results for the six GCs are used to validate the parallax bias correction as a
function of magnitude, color, and ecliptic latitude and to determine the
constant used to convert internal uncertainties to external ones. RESULTS. The
angular covariance at zero separation is 106.2 muas^2, yielding a minimum
uncertainty for EDR3 parallaxes of 10.3 muas for individual stars. That value
can be only slightly reduced for GCs after considering the behavior of the
angular covariance of the parallaxes for small separations. The Lindegren et
al. parallax bias correction works quite well, except for the brighter
magnitudes, suggesting improvements may be possible there. The value of k is
1.1-1.7 and depends on G. Stars with moderately large values of RUWE can still
provide useful parallaxes albeit with larger values of k. We give accurate and
precise Gaia EDR3 distances to the six GCs and for the specific case of 47 Tuc
we are able to beat the angular covariance limit and derive a high-precision
distance of 4.53+-0.06 kpc. Finally, a recipe for the derivation of distances
to stars and stellar clusters using Gaia EDR3 parallaxes is given. [ABRIDGED]

Instrumentation and Methods for Astrophysics;Astrophysics of Galaxies;Solar and Stellar Astrophysics