  Red supergiant (RSGs) are cool massive stars in a late phase of their
evolution when the stellar envelope becomes fully convective. They are the
brightest stars in the universe at infrared light and can be detected in
galaxies far beyond the Local Group, allowing for accurate determination of
chemical composition of galaxies. The study of their physical properties is
extremely important for various phenomena including the final fate of massive
stars as type II supernovae and gravitational wave progenitors. We explore the
well-studied nearby young stellar cluster chi Per. Using Gaia EDR3 data, we
find the distance of the cluster (d = 2.260+-0.020 kpc). We then investigate
the variability of the convection-related surface structure as a source for
parallax measurement uncertainty. We use state-of-the-art 3D radiative
hydrodynamics simulations with CO5BOLD and the post-processing radiative
transfer code OPTIM3D to compute intensity maps in the Gaia G photometric
system. We calculate the variabiltiy, as a function of time, of the
intensity-weighted mean from the synthetic maps. We then select the RSG stars
in the cluster and compare their uncertainty on parallaxes to the predictions
of photocentre displacements. The synthetic maps of RSG show extremely
irregular and temporal variable surfaces due to convection-related dynamics.
Consequentially, the position of the photo-center varies during Gaia
measurements between 0.033 and 0.130 AU (up to 5% of the corresponding
simulation stellar radius). We argue that the variability of the
convection-related surface structures accounts for a substantial part of the
Gaia EDR3 parallax error of the RSG sample. We suggest that the variation of
the uncertainty on Gaia parallax could be exploited quantitatively using
appropriate 3D simulations to extract, in a unique way, important information
about the stellar dynamics and parameters of RSG stars.
