Planetary nebulae (PNe) are a brief but important phase of stellar evolution. The study of Galactic PNe has historically been hampered by uncertain distances, but the parallaxes of PN central stars (CSPNe) measured by Gaia are improving the situation. Gaia's Early Data Release 3 (EDR3) offers higher astrometric precision and greater completeness compared to previous releases. Taking advantage of these improvements requires that the CSPNe in the catalogue be accurately identified. We applied our automated technique based on the likelihood ratio method to cross-match known PNe with sources in Gaia EDR3, using an empirically derived position and colour distribution to score candidate matches. We present a catalogue of over 2000 sources in Gaia EDR3 that our method has identified as likely CSPNe or compact nebula detections. We show how the more precise parallaxes of these sources compare to previous PN statistical distances and introduce an approach to combining them to produce tighter distance constraints. We also discuss Gaia's handling of close companions and bright nebulae. Gaia is unlocking new avenues for the study of PNe. The catalogue presented here will remain valid for the upcoming Gaia Data Release 3 (DR3) and thus provide a valuable resource for years to come.