  Gaia DR3 parallaxes are used to calibrate preliminary period--luminosity
relations of O-rich Mira variables in the 2MASS $J$, $H$ and $K_s$ bands using
a probabilistic model accounting for variations in the parallax zeropoint and
underestimation of the parallax uncertainties. The derived relations are
compared to those measured for the Large and Small Magellanic Clouds, the
Sagittarius dwarf spheroidal galaxy, globular cluster members and the subset of
Milky Way Mira variables with VLBI parallaxes. The Milky Way linear $JHK_s$
relations are slightly steeper and thus fainter at short period than the
corresponding LMC relations suggesting population effects in the near-infrared
are perhaps larger than previous observational works have claimed. Models of
the Gaia astrometry for the Mira variables suggest that, despite the intrinsic
photocentre wobble and use of mean photometry in the astrometric solution of
the current data reduction, the recovered parallaxes should be on average
unbiased but with underestimated uncertainties for the nearest stars. The
recommended Gaia EDR3 parallax zeropoint corrections evaluated at
$\nu_\mathrm{eff}=1.25\,\mu\mathrm{m}^{-1}$ require minimal
($\lesssim5\,\mu\mathrm{as}$) corrections for redder five-parameter sources,
but over-correct the parallaxes for redder six-parameter sources, and the
parallax uncertainties are underestimated, at most by a factor $\sim1.6$ at
$G\approx12.5\,\mathrm{mag}$. The derived period--luminosity relations are used
as anchors for the Mira variables in the Type Ia host galaxy NGC 1559 to find
$H_0=(73.7\pm4.4)\,\mathrm{km\,s}^{-1}\mathrm{Mpc}^{-1}$.
