We present the WFC3 Infrared Spectroscopic Parallel (WISP) Survey. WISP is obtaining slitless, near-infrared grism spectroscopy of ~90 independent, high-latitude fields by observing in the pure-parallel mode with the Wide Field Camera Three on the Hubble Space Telescope for a total of ~250 orbits. Spectra are obtained with the G 102 ( Delta *l = 0.8-1.17 Delta *mm, R ~210) and G 141 grisms ( Delta *l = 1.11-1.67 Delta *mm, R ~130), together with direct imaging in the J and H bands (F110W and F140W, respectively). In the present paper, we present the first results from 19 WISP fields, covering approximately 63 arcmin2. For typical exposure times (~6400 s in G 102 and ~2700 s in G 141), we reach 5 Delta *s detection limits for emission lines of f ~ 5 X 10--17 erg s--1 cm--2 for compact objects. Typical direct imaging 5 Delta *s limits are 26.3 and 26.1 mag. (AB) in F110W and F140W, respectively. Restricting ourselves to the lines measured with the highest confidence, we present a list of 328 emission lines, in 229 objects, in a redshift range 0.3 < z < 3. The single-line emitters are likely to be a mix of H Delta *a and O III5007,4959 A, with H Delta *a predominating. The overall surface density of high-confidence emission-line objects in our sample is approximately 4 per arcmin2. These first fields show high equivalent width sources, active galactic nucleus, and post-starburst galaxies. The median observed star formation rate (SFR) of our H Delta *a-selected sample is 4 M yr--1. At intermediate redshifts, we detect emission lines in galaxies as faint as H 140 ~ 25, or MR < --19, and are sensitive to SFRs down to less than 1 M yr--1. The slitless grisms on WFC3 provide a unique opportunity to study the spectral properties of galaxies much fainter than L* at the peak of the galaxy assembly epoch.