Using the recently upgraded Long Baseline Array, we have measured the trigonometric parallax of PSR J0437-4715 to better than 1% precision, the most precise pulsar distance determination made to date. Comparing this VLBI distance measurement to the kinematic distance obtained from pulsar timing, which is calculated from the pulsar's proper motion and apparent rate of change of orbital period, gives a precise limit on the unmodeled relative acceleration between the solar system and PSR J0437-4715, which can be used in a variety of applications. First, it shows that Newton's gravitational constant G is stable with time (Ġ/G = (-5 +/- 26) × 10-13 yr-1, 95% confidence). Second, if a stochastic gravitational wave background existed at the currently quoted limit, this null result would fail ~50% of the time. Third, it excludes Jupiter-mass planets within 226 AU of the Sun in 50% of the sky (95% confidence). Finally, the ~1% agreement of the parallax and orbital period derivative distances provides a fundamental confirmation of the parallax distance method on which all astronomical distances are based.