  Context. Asymptotic Giant Branch stars are characterised by complex stellar
surface dynamics that affect the measurements and amplify the uncertainties on
stellar parameters. The resulting motion of the stellar photo-center could have
adverse effects on the parallax determination with Gaia. Aims. We explore the
impact of the convection-related surface structure in AGBs on the photocentric
variability. We quantify these effects to characterise the observed parallax
errors and estimate fundamental stellar parameters and dynamical properties.
Methods. We use 3D radiative-hydrodynamics simulations of convection with
CO5BOLD and the post-processing radiative transfer code Optim3D to compute
intensity maps in the Gaia G band. From those maps, we calculate the
intensity-weighted mean of all emitting points tiling the visible stellar
surface and evaluate its motion as a function of time. We extract the parallax
error from Gaia DR2 for a sample of semiregular variables in the solar
neighbourhood and compare it to the synthetic predictions of photo-center
displacements. Results. AGB stars show a complex surface morphology
characterised by the presence of few large scale long-lived convective cells
accompanied by short-lived and small scale structures. As a consequence, the
position of the photo-center displays temporal excursions between 0.077 to
0.198 AU (5 to 11% of the corresponding stellar radius), depending on the
simulation considered. We show that the convection-related variability accounts
for a substantial part to the Gaia DR2 parallax error of our sample of
semiregular variables. Finally, we put in evidence for a correlation between
the mean photo-center displacement and the stellar fundamental parameters:
surface gravity, radius, and pulsation. We denote that parallax variations
could be exploited quantitatively using appropriate RHD simulations
corresponding to the observed star.
