  Relativistic models of light propagation adopted for high-precision
astrometry are based on the parametrised post-Newtonian formalism, which
provides a framework for examining the effects of a hypothetical violation of
general relativity on astrometric data. Astrometric observations are strongly
affected by the post-Newtonian parameter $\gamma$ describing the strength of
gravitational light deflection. We study both analytically and numerically how
a deviation in the PPN parameter $\gamma$ from unity, which is the value
predicted by general relativity, affects the parallax estimations in Gaia-like
astrometry. Changes in the observable quantities produced by a small variation
in PPN $\gamma$ were calculated analytically. We then considered how such
variations of the observables are reflected in the parallax estimations, and we
performed numerical simulations to check the theoretical predictions. A
variation in the PPN $\gamma$ results in a global shift of parallaxes and we
present a formula describing the parallax bias in terms of the satellite
barycentric distance, the angle between the spin axis and the direction to the
Sun, and the PPN $\gamma$ uncertainty. Numerical simulations of the astrometric
solutions confirm the theoretical result. The up-to-date estimation of PPN
$\gamma$ suggests that a corresponding contribution to the Gaia parallax zero
point unlikely exceeds 0.2 $\mu$as. The numerical simulations indicate that the
parallax shift is strongly dependent on ecliptic latitude. It is argued that
this effect is due to an asymmetry in the Gaia scanning law and this conclusion
is fully validated by additional simulations with a reversed direction of the
precession of the spin axis around the direction to the Sun.
