Gaia EDR3 has provided unprecedented data that has generated a great deal of interest in the astrophysical community, even though systematics affect the reported parallaxes at the level of similar to 10 mu as. 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 was done via a grid-based modelling method, where global asteroseismic observables, and 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. (2021a) 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.