We have used the Habitable Zone Planet Finder (HPF) to gather high-resolution, high signal-to-noise near-infrared spectra of 13 field red horizontal branch (RHB) stars, one open cluster giant, and one very metal-poor halo red giant. The HPF spectra cover the 0.81-1.28 mu m wavelength range of the zyJ bands, partially filling the gap between the optical (0.4-1.0 mu m) and infrared (1.5-2.4 mu m) spectra already available for the program stars. We derive abundances of 17 species from LTE-based computations involving equivalent widths and spectrum syntheses, and estimate abundance corrections for the species that are most affected by departures from LTE in RHB stars. Generally good agreement is found between HPF-based metallicities and abundance ratios and those from the optical and infrared spectral regions. Light element transitions dominate the HPF spectra of these red giants, and HPF data can be used to derive abundances from species with poor or no representation in optical spectra (e.g., C i, P i, S i, K i). Attention is drawn to the HPF abundances in two field solar-metallicity RHB stars of special interest: one with an extreme carbon isotope ratio, and one with a rare, very large lithium content. The latter star is unique in our sample in exhibiting very strong He i 10830 A absorption. The abundances of the open cluster giant concur with those derived from other wavelength regions. Detections of C i and S i in HD 122563 are reported, yielding the lowest metallicity determination of [S/Fe] from more than one multiplet.