Gaia measures the five astrometric parameters for stars in the Milky Way, but only four of them (positions and proper motion, but not distance) are well measured beyond a few kpc from the Sun. Modern spectroscopic surveys such as APOGEE cover a large area of the Milky Way disc 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+/-2.0{mu}as when modelling it as a global constant, but also train a multivariate zero-point offset model that depends on G, G_BP_-G_RP_ colour, and T_eff_ and that can be applied to all ~58 million stars in Gaia DR2 within APOGEE's colour-magnitude range and within APOGEE's sky footprint. Our spectro-photometric distances are more precise than Gaia at distances >~2kpc from the Sun. We release a catalogue of spectro-photometric distances for the entire APOGEE DR14 data set which covers Galactocentric radii 2kpc~