  This paper uses the multi-epoch astrometry from the Wide-field Infrared
Survey Explorer (WISE) to demonstrate a method to measure proper motions and
trigonometric parallaxes with precisions of $\sim$4 mas yr$^{-1}$ and $\sim$7
mas, respectively, for low-mass stars and brown dwarfs. This method relies on
WISE single exposures (Level 1b frames) and a Markov Chain Monte Carlo method.
The limitations of Gaia in observing very low-mass stars and brown dwarfs are
discussed, and it is shown that WISE will be able to measure astrometry past
the 95% completeness limit and magnitude limit of Gaia (L, T, and Y dwarfs
fainter than $G\approx19$ and $G=21$, respectively). This method is applied to
WISE data of 20 nearby ($\lesssim17$ pc) dwarfs with spectral types between
M6-Y2 and previously measured trigonometric parallaxes. Also provided are WISE
astrometric measurements for 23 additional low-mass dwarfs with spectral types
between M6-T7 and estimated photometric distances $<17$ pc. Only nine of these
objects contain parallaxes within Gaia Data Release 2.
