We present results from near-infrared spectroscopy of 26 emission-line galaxies at z ∼ 2.2 and z ∼ 1.5 obtained with the Folded-port InfraRed Echellette (FIRE) spectrometer on the 6.5 m Magellan Baade telescope. The sample was selected from the WFC3 Infrared Spectroscopic Parallels survey, which uses the near-infrared grism of the Hubble Space Telescope Wide Field Camera 3 (WFC3) to detect emission-line galaxies over 0.3 ≲ z ≲ 2.3. Our FIRE follow-up spectroscopy (R ∼ 5000) over 1.0-2.5 μm permits detailed measurements of the physical properties of the z ∼ 2 emission-line galaxies. Dust-corrected star formation rates for the sample range from ∼5-100 MȮ yr-1 with a mean of 29 MȮ yr-1. We derive a median metallicity for the sample of 12 + log(O/H) = 8.34 or ∼0.45 ZȮ. The estimated stellar masses range from ∼108.5-109.5 MȮ, and a clear positive correlation between metallicity and stellar mass is observed. The average ionization parameter measured for the sample, log U ≈ -2.5, is significantly higher than what is found for most star-forming galaxies in the local universe, but similar to the values found for other star-forming galaxies at high redshift. We derive composite spectra from the FIRE sample, from which we measure typical nebular electron densities of ∼100-400 cm-3. Based on the location of the galaxies and composite spectra on diagnostic diagrams, we do not find evidence for significant active galactic nucleus activity in the sample. Most of the galaxies, as well as the composites, are offset diagram toward higher [O III]/Hβ at a given [N II]/Hα, in agreement with other observations of z ≳ 1 star-forming galaxies, but composite spectra derived from the sample do not show an appreciable offset from the local star-forming sequence on the [O III]/Hβ versus [S II]/Hα diagram. We infer a high nitrogen-to-oxygen abundance ratio from the composite spectrum, which may contribute to the offset of the high-redshift galaxies from the local star-forming sequence in the [O III]/Hβ versus [N II]/Hα diagram. We speculate that the elevated nitrogen abundance could result from substantial numbers of Wolf-Rayet stars in starbursting galaxies at z ∼ 2. © 2014. The American Astronomical Society. All rights reserved.