The detection of radio pulsars within the central few parsecs of the Galaxy would provide a unique probe of the gravitational and magneto-ionic environments in the Galactic Center (GC) and, if close enough to Sgr A*, precise tests of general relativity in the strong-field regime. While it is difficult to find pulsars at radio wavelengths because of interstellar scattering, the payoff from detailed timing of pulsars in the GC warrants a concerted effort. To motivate pulsar surveys and help define search parameters for them, we constrain the pulsar number and spatial distribution using a wide range of multiwavelength measurements. These include the five known radio pulsars within 15 arcmin of Sgr A*, radio and gamma-ray measurements of diffuse emission, non-detections in high frequency pulsar surveys of the central parsec, a catalog of radio point sources from an imaging survey, infrared observations of massive star populations in the central few parsecs, candidate pulsar wind nebulae in the inner 20 pc and estimates of the core-collapse supernova rate based on X-ray measurements. We find that under current observational constraints, the inner parsec of the Galaxy could harbor as many as ~10^3 active radio pulsars that are beamed towards Earth. Such a large population would distort the low-frequency measurements of both the intrinsic spectrum of Sgr A* and the free-free absorption along the line of sight of Sgr A*. ; Comment: 17 pages, 4 figures, 4 tables, Published in ApJ, Updated to match published version