With contemporary infrared spectroscopic surveys like APO Galactic Evolution Experiment (APOGEE), red-giant stars can be observed to distances and extinctions at which Gaia parallaxes are not highly informative. Yet the combination of effective temperature, surface gravity, composition, and age-all accessible through spectroscopy - determines a giant's luminosity. Therefore spectroscopy plus photometry should enable precise spectrophotometric distance estimates. Here we use the overlap of APOGEE, Gaia, the Two Micron All Sky Survey (2MASS), and the Wide-field Infrared Survey Explorer (WISE) to train a data-driven model to predict parallaxes for red-giant branch stars with 0<logg=<2.2 (more luminous than the red clump). We employ (the exponentiation of) a linear function of APOGEE spectral pixel intensities and multiband photometry to predict parallax spectrophotometrically. The model training involves no logarithms or inverses of the Gaia parallaxes, and needs no cut on the Gaia parallax signal-to-noise ratio. It includes an L1 regularization to zero out the contributions of uninformative pixels. The training is performed with leave-out subsamples such that no star's astrometry is used even indirectly in its spectrophotometric parallax estimate. The model implicitly performs a reddening and extinction correction in its parallax prediction, without any explicit dust model. We assign to each star in the sample a new spectrophotometric parallax estimate; these parallaxes have uncertainties of less than 15%, depending on data quality, which is more precise than the Gaia parallax for the vast majority of targets, and certainly any stars more than a few kiloparsec distance. We obtain 10% distance estimates out to heliocentric distances of 20 kpc, and make global maps of the Milky Way's disk.