A body in solar orbit beyond the Kuiper Belt exhibits an annual parallax that exceeds its apparent proper motion by up to many orders of magnitude. Apparent motion of this body along the parallactic ellipse will deflect the angular position of background stars due to astrometric microlensing (induced parallax). By synoptically sampling the astrometric position of background stars over the entire sky, constraints on the existence (and basic properties) of a massive nearby body may be inferred. With a simple simulation, we estimate the signal-to-noise ratio for detecting such a body-as a function of mass, heliocentric distance, and ecliptic latitude-using the anticipated sensitivity and temporal cadences from Gaia (launch date 2011). A Jupiter-mass (M$_J$) object at 2000 AU is detectable by Gaia over the whole sky above 5 σ, with even stronger constraints if it lies near the ecliptic plane. Hypotheses for the mass (åisebox-0.5ex 3M$_J$), distance (i̊sebox-0.5ex 20,000 AU), and location of the proposed perturber (``Planet X’'), which gives rise to long- period comets, may be testable.