  Gaia EDR3 has provided unprecedented data that generate a lot of interest in
the astrophysical community, despite the fact that systematics affect the
reported parallaxes at the level of ~ 10 muas. Independent distance
measurements are available from asteroseismology of red-giant stars with
measurable parallaxes, whose magnitude and colour ranges more closely reflect
those of other stars of interest. In this paper, we determine distances to
nearly 12,500 red-giant branch and red clump stars observed by Kepler, K2, and
TESS. This is done via a grid-based modelling method, where global
asteroseismic observables, constraints on the photospheric chemical
composition, and on the unreddened photometry are used as observational inputs.
This large catalogue of asteroseismic distances allows us to provide a first
comparison with Gaia EDR3 parallaxes. Offset values estimated with
asteroseismology show no clear trend with ecliptic latitude or magnitude, and
the trend whereby they increase (in absolute terms) as we move towards redder
colours is dominated by the brightest stars. The correction model proposed by
Lindegren et al. (2021) is not suitable for all the fields considered in this
study. We find a good agreement between asteroseismic results and model
predictions of the red clump magnitude. We discuss possible trends with the
Gaia scan law statistics, and show that two magnitude regimes exist where
either asteroseismology or Gaia provides the best precision in parallax.
