We have imaged the classical Seyfert galaxy NGC 4151 with the VLBA at wavelengths of 6 and 18 cm, achieving resolutions of 2 and 5 mas (0.16 and 0.40 pc), respectively. At 18 cm, four radio components, spread over 05 (40 pc) are detected. The easternmost pair comprises a bright component elongated in position angle 20° (component E) and an extension to the northeast (component F) which curves into position angle 75°, similar to that of the 35 (280 pc) scale radio jet and the narrow-line region. At the higher resolution achieved at 6 cm, component E is found to be a linear radio source with a length of ~13 mas (1.0 pc). This source has a length/width ratio of 4, and therefore fulfills one of the classical criteria for a radio jet, but its radio luminosity is only ~1038 ergs s-1, several orders of magnitude less than the parsec-scale jets in radio galaxies. It is also misaligned by 55° from the arcsecond-scale radio jet. Possible reasons for the misalignment include deflection by the inner narrow-line region, buoyancy forces, and a change in the plane of the accretion disk presumably responsible for jet collimation. We also discuss an alternative interpretation, in which component E is a disk or torus viewed edge-on and emitting flat-spectrum, possibly thermal, radio emission. Component E appears to be the only part of the radio emission that contains a significant flat-spectrum component, and we favor identification of the central, brightest, unresolved subcomponent of E as the ultraviolet (UV) and optical nucleus. In order to reconcile the much smaller column density of H I toward the nucleus found by Lyman absorption than by 21 cm absorption measurements, we argue that a 0.01 pc thick gas disk surrounds the nucleus and is ionized out to a radius of 2 pc. The large 21 cm absorption column observed then results from off-nuclear radio components shining through the outer, neutral part of this disk. The flat spectrum of the nuclear radio source may indicate synchrotron self-absorption or free-free absorption by the inner, ionized part of the accretion disk. Interestingly, the apparent nuclear source and a radio subcomponent 0.2 pc to the southeast align precisely perpendicular to the arcsec-scale radio jet, suggesting that they may outline the large-scale structure of the accretion disk responsible for jet collimation. Comparison of the 18 cm image with the European VLBI network image acquired by Harrison et al. in 1984 provides upper limits of 0.14c and 0.25c for the apparent speeds of the radio components at distances of 7 and 36 pc, respectively, from the galaxy nucleus.