We present a catalogue of 362 million stellar parameters, distances, and extinctions derived from Gaia's Early Data Release (EDR3) cross-matched with the photometric catalogues of Pan-STARRS1, SkyMapper, 2MASS, and All WISE. The higher precision of the Gaia EDR3 data, combined with the broad wavelength coverage of the additional photometric surveys and the new stellar-density priors of the StarHorse code, allows us to substantially improve the accuracy and precision over previous photo-astrometric stellar-parameter estimates. At magnitude G = 14 (17), our typical precisions amount to 3% (15%) in distance, 0.13 mag (0.15 mag) in V-band extinction, and 140 K (180 K) in effective temperature. Our results are validated by comparisons with open clusters, as well as with asteroseismic and spectroscopic measurements, indicating systematic errors smaller than the nominal uncertainties for the vast majority of objects. We also provide distance- and extinction-corrected colour-magnitude diagrams, extinction maps, and extensive stellar density maps that reveal detailed substructures in the Milky Way and beyond. The new density maps now probe a much greater volume, extending to regions beyond the Galactic bar and to Local Group galaxies, with a larger total number density. We publish our results through an ADQL query interface (gaia . aip . de) as well as via tables containing approximations of the full posterior distributions. Our multi-wavelength approach and the deep magnitude limit render our results useful also beyond the next Gaia release, DR3.