Context. It has been shown that convection in red supergiant stars (RSG) gives rise to large granules that cause surface inhomogeneities and shock waves in the photosphere. The resulting motion of the photocentre (on time scales ranging from months to years) could possibly have adverse effects on the parallax determination with Gaia. Aims. We explore the impact of the granulation on the photocentric and photometric variability. We quantify these effects in order to better characterise the error that could possibly alter the parallax. Methods. We use 3D radiative-hydrodynamics (RHD) simulations of convection with CO5BOLD and the post-processing radiative transfer code Optim3D to compute intensity maps and spectra in the Gaia G band 325-1030 nm. Results. We provide astrometric and photometric predictions from 3D simulations of RSGs that are used to evaluate the possible degradation of the astrometric parameters of evolved stars derived by Gaia. We show in particular from RHD simulations that a supergiant like Betelgeuse exhibits a photocentric noise characterised by a standard deviation of the order of 0.1 AU. The number of bright giant and supergiant stars whose Gaia parallaxes will be altered by the photocentric noise ranges from a few tens to several thousands, depending on the poorly known relation between the size of the convective cells and the atmospheric pressure scale height of supergiants, and to a lower extent, on the adopted prescription for galactic extinction. In the worst situation, the degradation of the astrometric fit caused by this photocentric noise will be noticeable up to about 5 kpc for the brightest supergiants. Moreover, parallaxes of Betelgeuse-like supergiants are affected by an error of the order of a few percents. We also show that the photocentric noise, as predicted by the 3D simulation, does account for a substantial part of the supplementary "cosmic noise" that affects Hipparcos measurements of Betelgeuse and Antares.