Context. 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, they are systematically offset from the expected distribution around zero, by a few tens of microarcseconds. Aims. We attempt to map the main dependences of the parallax bias in EDR3. In principle, this could provide a recipe for correcting the EDR3 parallaxes. Methods. Quasars provide the most direct way for estimating the parallax bias for faint sources. In order to extend this to brighter sources and a broader range of colours, we used differential methods based on physical pairs (binaries) and sources in the Large Magellanic Cloud. The functional forms of the dependences were explored by mapping the systematic differences between EDR3 and DR2 parallaxes. Results. 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 dependences 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.