  Using the nine-year radio-pulsar timing data set from the North American
Nanohertz Observatory for Gravitational Waves (NANOGrav), collected at Arecibo
Observatory and the Green Bank Telescope, we have measured the positions,
proper motions, and parallaxes for 37 millisecond pulsars. We report twelve
significant parallax measurements and distance measurements, and eighteen lower
limits on distance. We compare these measurements to distances predicted by the
NE2001 interstellar electron density model and find them to be in general
agreement. We use measured orbital-decay rates and spin-down rates to confirm
two of the parallax distances and to place distance upper limits on other
sources; these distance limits agree with the parallax distances with one
exception, PSR J1024-0719, which we discuss at length. Using the proper motions
of the 37 NANOGrav pulsars in combination with other published measurements, we
calculate the velocity dispersion of the millisecond pulsar population in
Galactocentric coordinates. We find the radial, azimuthal, and perpendicular
dispersions to be 46, 40, and 24 km s-1, respectively, in a model that allows
for high-velocity outliers; or 81, 58, and 62 km s-1 for the full population.
These velocity dispersions are far smaller than those of the canonical pulsar
population, and are similar to older Galactic disk populations. This suggests
that millisecond pulsar velocities are largely attributable to their being an
old population rather than being artifacts of their birth and evolution as
neutron star binary systems. The components of these velocity dispersions
follow similar proportions to other Galactic populations, suggesting that our
results are not biased by selection effects.
