  We present absolute parallaxes and relative proper motions for the central
stars of the planetary nebulae NGC 6853 (The Dumbbell), NGC 7293 (The Helix),
Abell 31, and DeHt 5. This paper details our reduction and analysis using DeHt
5 as an example. We obtain these planetary nebula nuclei (PNNi) parallaxes with
astrometric data from Fine Guidance Sensors FGS 1R and FGS 3, white-light
interferometers on the Hubble Space Telescope (HST). Proper motions, spectral
classifications and VJHKT_2M and DDO51 photometry of the stars comprising the
astrometric reference frames provide spectrophotometric estimates of reference
star absolute parallaxes. Introducing these into our model as observations with
error, we determine absolute parallaxes for each PNN. Weighted averaging with
previous independent parallax measurements yields an average parallax
precision, \sigma_{\pi}/\pi = 5 %. Derived distances are: d_{NGC
6853}=405^{+28}_{-25}pc, d_{NGC 7293}=216^{+14}_{-12} pc, d_{Abell 31} =
621^{+91}_{-70} pc, and d_{DeHt 5} = 345^{+19}_{-17} pc. These PNNi distances
are all smaller than previously derived from spectroscopic analyses of the
central stars. Derived absolute magnitudes and previously measured effective
temperatures permit estimates of PNNi radii, through both the Stefan-Boltzmann
relation and Eddington fluxes. Comparing absolute magnitudes with post-AGB
models provides mass estimates. Masses cluster around 0.57 M(sun), close to the
peak of the white dwarf mass distribution. Adding a few more PNNi with
well-determined distances and masses, we compare all the PNNi with cooler white
dwarfs of similar mass, and confirm, as expected, that PNNi have larger radii
than white dwarfs that have reached their final cooling tracks. (Abridged)
