We observed 17 nearby low-luminosity FR I radio galaxies using the NRAO Very Long Baseline Array (VLBA) at 1.67 GHz, as part of a multiwavelength study of a complete sample of 21 sources Selected by radio flux density from the Uppsala General Catalogue of Galaxies. We detected radio emission from all 17 galaxies. At a FWHM resolution of approximate to 10 x 4 mas, five galaxies show only an unresolved radio core, 10 galaxies show core-jet structures, and two galaxies show twin-jet structures. Comparing these VLBA images with images previously obtained with the NRAO VLA, we find that all detected VLBA jets are well aligned on parsec scales with the VLA jets on kiloparsec scales and that the jet-to-counterjet surface brightness ratios, or the sidedness, decreases systematically with increasing distance along the jet. We attribute the sidedness to the Doppler boosting effect and its decline to the deceleration of the jets. We show that a distribution of Lorentz factor centered near Gamma = 5 can reproduce our VLBA detection statistics for core, core-jet, and twin-jet sources. We also note that the luminosity per unit length, L-j, of the VLBA jets drops quickly with distance, r, along the jet, approximately as L-j proportional to r(-2.0). We discuss three different mechanisms to explain this jet fading: (1) the decrease of Doppler boosting due to jet deceleration, (2) synchrotron losses, and (3) expansion losses in constant velocity but adiabatically spreading jets. Mechanisms (1) and (2) are inconsistent with the observations, while mechanism (3) is consistent with the observations provided the magnetic field lines in the jets are aligned perpendicular to the jet axis. This implies that the deceleration of the jets required by the unified scheme does not occur on the tens of parsec scales but must occur on larger scales.