  Gaia Early Data Release 3 (Gaia EDR3) gives trigonometric parallaxes for
nearly 1.5 billion sources. Inspection of the EDR3 data for sources identified
as quasars reveals that their parallaxes are biased, that is systematically
offset from the expected distribution around zero, by a few tens of
microarcsec. We attempt to map the main dependencies of the parallax bias in
EDR3. In principle this could provide a recipe for correcting the EDR3
parallaxes. For faint sources the quasars provide the most direct way to
estimate parallax bias. In order to extend this to brighter sources and a
broader range of colours, we use differential methods based on physical pairs
(binaries) and sources in the Large Magellanic Cloud. The functional forms of
the dependencies are explored by mapping the systematic differences between
EDR3 and DR2 parallaxes. The parallax bias is found to depend in a non-trivial
way on (at least) the magnitude, colour, and ecliptic latitude of the source.
Different dependencies apply to the five- and six-parameter solutions in EDR3.
While it is not possible to derive a definitive recipe for the parallax
correction, we give tentative expressions to be used at the researcher's
discretion and point out some possible paths towards future improvements.
