  Astrometry can bring powerful constraints to bear on a variety of scientific
questions about neutron stars, including their origins, astrophysics,
evolution, and environments. Using phase-referenced observations at the VLBA,
in conjunction with pulsar gating and in-beam calibration, we have measured the
parallaxes and proper motions for 14 pulsars. The smallest measured parallax in
our sample is 0.13+-0.02 mas for PSR B1541+09, which has a most probable
distance of 7.2+1.3-1.1 kpc. We detail our methods, including initial VLA
surveys to select candidates and find in-beam calibrators, VLBA
phase-referencing, pulsar gating, calibration, and data reduction. The use of
the bootstrap method to estimate astrometric uncertainties in the presence of
unmodeled systematic errors is also described. Based on our new
model-independent estimates for distance and transverse velocity, we
investigate the kinematics and birth sites of the pulsars and revisit models of
the Galactic electron density distribution. We find that young pulsars are
moving away from the Galactic plane, as expected, and that age estimates from
kinematics and pulsar spindown are generally in agreement, with certain notable
exceptions. Given its present trajectory, the pulsar B2045-16 was plausibly
born in the open cluster NGC 6604. For several high-latitude pulsars, the
NE2001 electron density model underestimates the parallax distances by a factor
of two, while in others the estimates agree with or are larger than the
parallax distances, suggesting that the interstellar medium is irregular on
relevant length scales. The VLBA astrometric results for the recycled pulsar
J1713+0747 are consistent with two independent estimates from pulse timing,
enabling a consistency check between the different reference frames.
