Active galaxies are thought to be both fueled and obscured by neutral gas removed from the host galaxy and funneled into a central accretion disk. We performed a VLA imaging survey of 21 cm absorption in Seyfert and starburst nuclei to study the neutral gas in the near-nuclear environment. With the exception of NGC 4151, the absorbing gas traces 100 pc scale, rotating disks aligned with the outer galaxy disk. These disks appear to be rich in atomic gas relative to nuclear disks in non-active spirals. We find no strong evidence for rapid in-fall or out-flow of neutral hydrogen, but our limits on the mass infall rates are compatible with that required to feed a Seyfert nucleus. Among the galaxies surveyed here, neutral hydrogen absorption traces parsec-scale gas only in NGC 4151. Based on the kinematics of the absorption line, the disk symmetry axis appears to align with the radio jet axis rather than the outer galaxy axis. The most surprising result is that we detect no 21 cm absorption towards the central radio sources of the hidden Seyfert 1 nuclei Mrk 3, Mrk 348, and NGC 1068. Moreover, 21 cm absorption is commonly observed towards extended radio jet structure but appears to avoid central, compact radio sources in Seyfert nuclei. To explain these results, we propose that 21 cm absorption towards the nucleus is suppressed by either free-free absorption, excitation effects (i.e., enhanced spin temperature), or rapid motion in the obscuring gas. Ironically, the implications of these effects is that the obscuring disks must be small, typically not larger than a few tens of parsecs.