  Gaia measures the five astrometric parameters for stars in the Milky Way, but
only four of them (positions and proper motion, but not parallax) are well
measured beyond a few kpc from the Sun. Modern spectroscopic surveys such as
APOGEE cover a large area of the Milky Way disk and we can use the relation
between spectra and luminosity to determine distances to stars beyond Gaia's
parallax reach. Here, we design a deep neural network trained on stars in
common between Gaia and APOGEE that determines spectro-photometric distances to
APOGEE stars, while including a flexible model to calibrate parallax zero-point
biases in Gaia DR2. We determine the zero-point offset to be $-52.3 \pm 2.0uas$
when modeling it as a global constant, but also train a multivariate zero-point
offset model that depends on $G$, $G_{BP} - G_{RP}$ color, and $T_\mathrm{eff}$
and that can be applied to all 139 million stars in Gaia DR2 within APOGEE's
color--magnitude range. Our spectro-photometric distances are more precise than
Gaia at distances $\approx 2kpc$ from the Sun. We release a catalog of
spectro-photometric distances for the entire APOGEE DR14 data set which covers
Galactocentric radii $2kpc\lesssim R \lesssim19kpc$; $\approx 150,000$ stars
have <10% uncertainty, making this a powerful sample to study the
chemo-dynamical structure of the disk. We use this sample to map the mean
[Fe/H] and 15 abundance ratios [X/Fe] from the Galactic center to the edge of
the disk. Among many interesting trends, we find that the bulge and bar region
at $R \lesssim 5kpc$ clearly stands out in [Fe/H] and most abundance ratios.
