We present near-infrared emission line counts and luminosity functions from the Hubble Space Telescope Wide Field Camera 3 Infrared Spectroscopic Parallels (WISP) program for 29 fields (0.037 deg(2)) observed using both the G102 and G141 grism. Altogether we identify 1048 emission line galaxies with observed equivalent widths greater than 40 angstrom, 467 of which have multiple detected emission lines. We use simulations to correct for significant (>20%) incompleteness introduced in part by the non-dithered, non-rotated nature of the grism parallels. The WISP survey is sensitive to fainter flux levels ((3-5) x 10(-17) erg s(-1) cm(-2)) than the future space near-infrared grism missions aimed at baryonic acoustic oscillation cosmology ((1-4) x 10(-16) erg s(-1) cm(-2)), allowing us to probe the fainter emission line galaxies that the shallower future surveys may miss. Cumulative number counts of 0.7 < z < 1.5 galaxies reach 10,000 deg(-2) above an H alpha flux of 2 x 10(-16) erg s(-1) cm(-2). H alpha-emitting galaxies with comparable [O III] flux are roughly five times less common than galaxies with just H alpha emission at those flux levels. Galaxies with low H alpha/[O III] ratios are very rare at the brighter fluxes that future near-infrared grism surveys will probe; our survey finds no galaxies with H alpha/[O III] < 0.95 that have H alpha flux greater than 3 x 10(-16) erg s(-1) cm(-2). Our H alpha luminosity function contains a comparable number density of faint line emitters to that found by the Near IR Camera and Multi-Object Spectrometer near-infrared grism surveys, but significantly fewer (factors of 3-4 less) high-luminosity emitters. We also find that our high-redshift (z = 0.9-1.5) counts are in agreement with the high-redshift (z = 1.47) narrowband H alpha survey of HiZELS (Sobral et al.), while our lower redshift luminosity function (z = 0.3-0.9) falls slightly below their z = 0.84 result. The evolution in both the H alpha luminosity function from z = 0.3-1.5 and the [O III] luminosity function from z = 0.7-2.3 is almost entirely in the L-star parameter, which steadily increases with redshift over those ranges.