  CONTEXT. The Gaia EDR3 parallaxes constitute the most detailed and accurate
dataset that can be currently used to determine stellar distances in the solar
neighborhood. Nevertheless, there is still room for improvement in their
calibration and systematic effects can be further reduced in some
circumstances.
  AIMS. The aim of this paper is to determine an improved Gaia EDR3 parallax
bias as a function of magnitude, color, and ecliptic latitude using a single
method applied to stars in open clusters, globular clusters, the LMC, and the
SMC.
  METHODS. I study the behavior of the residuals or differences between the
individual (stellar) parallaxes and the group parallaxes, which are assumed
constant for the corresponding cluster or galaxy. This is done by first
applying the Lindegren et al. (2021) zero point and then calculating a new zero
point from the residuals of the first analysis.
  RESULTS. The Lindegren zero point shows very small residuals as a function of
magnitude between individual and group parallaxes for G < 13 but significant
ones for brighter stars, especially blue ones. The new zero point reduces those
residuals, especially in the 9.2 < G < 13 range. The k factor that is used to
convert from catalog parallax uncertainties to external uncertainties is small
(1.1-1.7) for 9.2 < G < 11 and G > 13, intermediate (1.7-2.0) for 11 < G < 13,
and large (>2.0) for G <9.2. Therefore, significant corrections are needed to
calculate distance uncertainties from Gaia EDR3 parallaxes for some stars.
There is still room for improvement if future analyses add information from
additional stellar clusters, especially for red stars with G < 11 and blue
stars with G < 9.2. I also calculate k for stars with RUWE values between 1.4
and 8.0 and for stars with 6-parameter solutions, allowing for a correct
estiimation of their uncertainties.
  [ABRIDGED]
