Context. We present a new methodology for the estimation of stellar atmospheric parameters from narrow- and intermediate-band photometry of the Javalambre Photometric Local Universe Survey (J-PLUS), and propose a method for target pre-selection of low-metallicity stars for follow-up spectroscopic studies. Photometric metallicity estimates for stars in the globular cluster M15 are determined using this method. Aims: By development of a neural-network-based photometry pipeline, we aim to produce estimates of effective temperature, Teff, and metallicity, [Fe/H], for a large subset of stars in the J-PLUS footprint. Methods: The Stellar Photometric Index Network Explorer, SPHINX, was developed to produce estimates of Teff and [Fe/H], after training on a combination of J-PLUS photometric inputs and synthetic magnitudes computed for medium-resolution (R 2000) spectra of the Sloan Digital Sky Survey. This methodology was applied to J-PLUS photometry of the globular cluster M15. Results: Effective temperature estimates made with J-PLUS Early Data Release photometry exhibit low scatter, σ(Teff) = 91 K, over the temperature range 4500 < Teff (K) < 8500. For stars from the J-PLUS First Data Release with 4500 < Teff (K) < 6200, 85 ± 3% of stars known to have [Fe/H] < -2.0 are recovered by SPHINX. A mean metallicity of [Fe/H] = - 2.32 ± 0.01, with a residual spread of 0.3 dex, is determined for M15 using J-PLUS photometry of 664 likely cluster members. Conclusions: We confirm the performance of SPHINX within the ranges specified, and verify its utility as a stand-alone tool for photometric estimation of effective temperature and metallicity, and for pre-selection of metal-poor spectroscopic targets.