Aims. We investigated the location of the optical emission with respect to the parsec-scale radio structure and attempted to identify whether the optical centroid is coincident with the radio core or a radio component of the jet.Methods. We used optical Gaia DR3 and 8 GHz ICRF3 positions and a model fitting of 15 GHz imaging by the MOJAVE VLBA survey for 422 sources, mostly blazars. We searched for possible associations between optical centroids and radio components. We also compared the astrometric and photometric properties of each category, looked for trends between the various source parameters, and discussed possible biases.Results. Sources can be separated into four categories based on whether their optical centroid (i) falls onto the radio core (category C), (ii) lies close to the base of the jet (category B), (iii) coincides with a radio component downstream in the jet (category J), or (iv) is not found to coincide with a detected radio component (category O). Due to a number of random and systematic errors, the number of sources falling into each category remains approximate but close to 32%, 36%, 22%, and 10%, respectively. The family of quasars, mostly flat spectrum radio quasars (FSRQs), clearly dominates the C and B categories, with their percentage decreasing from categories C to O. Conversely, the family of BL Lacs is spread over the four classes and dominates the category O. Radio galaxies are mainly in classes O and J, and the radio-loud narrow-line Seyfert 1 all belong to class C. An individual analysis of O sources shows that, despite the absence of a direct association with a specific radio component, the optical Gaia centroid is globally related to the radio VLBI jet structure.Conclusions. Our study shows that the spatial distribution of the optical counterparts seen by Gaia below the 0.2-arcsec scale around the radio cores appears in the vast majority of cases to be related to the AGN core or to its jet, with most of them located downstream in the jet. Those associated to the core also exhibit a bluer color index, suggesting a possible contribution of the accretion disk to the optical emission, while those associated to a radio knot in the jet appear redder and tend on average to be found in more polarized radio sources. Most BL Lacs have their optical emission coincident with the jet base or a knot in the jet, while sources with an optical emission on or close to the radio core are mostly FSRQs. Radio components associated with the optical centroid at the jet base or along the jet are mainly stationary or quasi-stationary features, with low apparent MOJAVE velocities. There are indications that the apparent proper motion of the Gaia centroids may be higher than the speed of the associated radio components, but the significance of this trend requires further investigation.