  As a step toward investigating the parsec-scale properties of faint
extragalactic radio sources, the Very Long Baseline Array (VLBA) was used at
5.0 GHz to obtain phase-referenced images of 76 sources in the NOAO Bootes
field. These 76 sources were selected from the FIRST catalog to have peak flux
densities above 10 mJy at 5 arcsec resolution and deconvolved major diameters
of less than 3 arcsec at 1.4 GHz. Fifty-seven of these faint radio sources were
identified with accretion-powered radio galaxies and quasars brighter than 25.5
mag in the optical I band. On VLA scales at 1.4 GHz, a measure of the
compactness of the faint sources (the ratio of the peak flux density from FIRST
to the integrated flux density from the NVSS catalog) spans the full range of
possibilites arising from source-resolution effects. Thirty of the faint radio
sources, or 39 +9/-7%, were detected with the VLBA at 5.0 GHz with peak flux
densities above 6 sigma ~2 mJy at 2 mas resolution. The VLBA detections occur
through the full range of compactness ratios. The stronger VLBA detections can
themselves serve as phase-reference calibrators, boding well for opening up
much of the radio sky to VLBA imaging. For the adopted cosmology, the VLBA
resolution correponds to 17 pc or finer. Most VLBA detections are unresolved or
slightly resolved but one is diffuse and five show either double or core-jet
structures; the properties of these latter six are discussed in detail. Three
VLBA detections are unidentified and fainter than 25.5 mag in the optical I
band; their properties are highlighted because they likely mark
optically-obscured active nuclei at high redshift.
