  The systematic offset of Gaia parallaxes has been widely reported with Gaia's
second data release, and it is expected to persist in future Gaia data. In
order to use Gaia parallaxes to infer distances to high precision, we develop a
hierarchical probabilistic model to determine the Gaia parallax zero point
offset along with the calibration of an empirical model for luminosity of red
clump stars by combining astrometric and photometric measurements. Using a
cross-matched sample of red clump stars from the Apache Point Observatory
Galactic Evolution Experiment (APOGEE) and Gaia Data Release 2 (DR2), we report
the parallax zero point offset in DR2 to be $ \varpi_0 = -48 \pm 1~\mu\text{as}
$. We infer the red clump absolute magnitude to be $ -1.622 \pm 0.004 $ in $
K_s $, $ 0.517 \pm 0.004 $ in $ G $, $ -1.019 \pm 0.004 $ in $ J $, and $
-1.516 \pm 0.004 $ in $ H $. The intrinsic scatter of the red clump is $ \sim
0.09 $ mag in $ J $, $ H $ and $ K_s $, or $ \sim 0.17 $ mag in $ G $. We
tailor our models to accommodate more complex analyses such as investigating
the variations of the parallax zero point with each source's observed
magnitude, observed colour, and sky position. In particular, we find
fluctuations of the zero point across the sky to be of order or less than a few
10s of $ \mu\text{as} $.
