Not yet reviewed by Pith; the record is open.

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]