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 2 M 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, σ π /π = 5%. Derived distances are: d N GC 6853 = 405 +28 −25 pc, d N GC 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. To obtain absolute magnitudes from these distances requires estimates of interstellar extinction. We average extinction measurements culled -2from the literature, from reddening based on PNNi intrinsic colors derived from model SEDs, and an assumption that each PNN experiences the same rate of extinction as a function of distance as do the reference stars nearest (in angular separation) to each central star. We also apply Lutz-Kelker bias corrections. The absolute magnitudes and 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 ⊙ , 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.Subject headings: astrometry -interferometry -planetary nebulae -stars: distances -stars: white dwarf -stars: masses Processing and Analysis Center/California Institute of Technology, funded by NASA and the NSF. This research has made use of the SIMBAD database and Aladin, both developed at CDS, Strasbourg, France; the NASA/IPAC Extragalactic Database (NED) which is operated by JPL, California Institute of Technology, under contract with the NASA; and NASA's Astrophysics Data System Abstract Service. We thank an anonymous referee for a careful reading and suggestions that improved the presentation.