We have used the Very Long Baseline Array to produce a high dynamic range image of the nucleus of NGC 6251 at 1.6 GHz and snapshot images at 5.0, 8.4, and 15.3 GHz to search for emission from a parsec-scale counterjet. Previous VLBI images at 1.6 GHz have set a lower limit for the jet/counterjet brightness ratio near the core at about 80:1, which is larger than expected, given the evidence that the radio axis is fairly close to the plane of the sky. A possible explanation is that the inner few parsecs of the counterjet are hidden by free-free absorption by ionized gas associated with an accretion disk or torus. This would be consistent with the nearly edge-on appearance of the arcsecond-scale dust disk seen in the center of NGC 6251 by Hubble Space Telescope (HST). We detect counterjet emission close to the core at 1.6 GHz, but not at the higher frequencies. Given that the optical depth of free-free absorption falls off more rapidly with increasing frequency than the optically thin synchrotron emission from a typical radio jet, this result implies that the absence of a detectable parsec-scale counterjet at high frequencies is not due to free-free absorption unless the density of ionized gas is extremely high and we have misidentified the core at 1.6 GHz. The most likely alternative is a large jet/counterjet brightness ratio caused by relativistic beaming, which in turn requires the inner radio axis to be closer to our line of sight than the orientation of the HST dust disk would suggest.