We examine the zero-point of parallaxes in the Gaia EDR3 (Early Data Release 3, Gaia Collaboration (2021a)), using stars in open and globular clusters. Our aim is to check for zero-point systematics between bright stars G < 12 (which includes some important distance scale calibrators) and faint stars G > 14, for which the parallax scale has been very well tied to the extragalactic frame using active galactic nuclei and quasars (AGN/QSOs) observed by Gaia (Lindegren et al. 2021). Cluster stars are distinguished from field stars using conservative spatial and proper motion cuts. The median parallax cluster stars fainter the G = 14 are compared with the parallaxes of bright stars (G < 14) to search for zero-point systematics. We confirm that the Lindegren et al. (2021) parallax corrections bring cluster stars into very good agreement over a wide range of magnitude and colour. We find small residual colour-dependent offsets for the bright stars (G < 11). Specifically, we find a median parallax offset of $\approx 10\, \mu$as between the reddest stars (BP - RP > 1) compared to those stars with colours similar to the AGN/QSOs (0.5 < BP - RP < 1) that serve as the primary zero-point calibrators for EDR3. These findings are similar to those found in other recent independent checks of the zero-point scale, and have significant implications for calibrating the extragalactic distance scale to stars in the Milky Way.