  (Abridged) A consequence of the Earth's motion with respect to the CMB is
that over a 10 year period it will travel a distance of ~800 AU. As first noted
by Kardashev in 1986, this baseline can be used to carry out astrometric
measurements of quasar parallaxes, so that only microarcsecond precision is
necessary to detect parallax shifts of objects at gigaparsec distances. Such
precision will soon be approached with the launch of the astrometric satellites
Gaia and SIM. We use a Fisher matrix formalism to investigate the constraints
that these and future missions may be able to place on the cosmological
distance scale and dark energy. We find that by observing around a million
quasars as planned, an extended 10 year Gaia mission could detect quasar
parallax shifts at the 2.8 sigma level and so measure the Hubble constant to
within 25 km/s. For the interferometer SIMLite, a Key Project using 2.4 % of
the total mission time to observe 750 quasars could detect the effect at the 2
sigma level. Gaia and a dedicated SIMLite only weakly constrain the presence of
a cosmological constant at the ~1 sigma levels. We also investigate future
mission concepts, such as an interferometer similar in scope and design to
NASA's Terrestrial Planet Finder. This could in principle measure the dark
energy parameters w_0 and w_a with high precision, yielding a Figure of Merit
larger than the stage IV experiments considered by the the Dark Energy Task
Force. Unlike perhaps all other probes of dark energy there appear to be no
obvious astrophysical sources of systematic error. There is however uncertainty
regarding the statistical errors. As well as measurement error, there will be
small additional contributions from image centroiding of variable sources,
quasar peculiar motions and weak microlensing by stars along the line of sight.
