We present very long baseline interferometry (VLBI) images of the core emission from a nearby bright FR II radio galaxy, Pictor A, revealing its parsec-scale jet structure and evolution for the first time. These data constitute a significant addition to our knowledge of powerful radio galaxies on the smallest scales, effectively doubling the number studied at this resolution. The jet, 14 h-1 pc in projected extent, is directed west of the core for the first 5 h-1 pc and then appears to bend approximately 40° to the north. Apparent motions for three of the five parsec-scale jet components have been estimated, 0.5 ± 0.4, 1.1 ± 0.5, and 0.4 ± 0.7 h-1c, indicating that subluminal motion is likely. No parsec-scale counterjet has been detected, allowing only lower limits on the jet-to-counterjet surface brightness ratio to be estimated. Two models, one describing the apparent 40° bend in the parsec-scale jet as an intrinsic deflection of the jet and one describing it as the effect of jet precession, may each be plausible and should be testable with future VLBI observations. By adopting the jet deflection model to describe the apparent 40° bend, we estimate that the Pictor A jet is initially inclined to our line of sight by less than 51°. Comparing this result with VLBI observations of Cygnus A suggests that, while the components in both jets are consistent with at least mildly relativistic speeds, the Pictor A jet lies significantly closer to our line of sight than the Cygnus A jet. This conclusion is consistent with both the parsec-scale radio structures and the kiloparsec-scale orientations of the host galaxies as well as the "unified model" interpretation of the optical spectra from these two objects.