Although an astrometric spacecraft sent out of the Solar System will get a much larger parallax than a near-Earth astrometric spacecraft ever will, it will also have the serious problem of sending its data back to Earth, simply because of its large distance.

GAIA will be at 0.01 AU or 1.5 million km, while to significantly improve on the Earth's orbit requires a distance of at least 10 AU, or 1.5 billion km. If one assumes that both satellites transmit their data with the same transmitter power, then the outer-Solar-System one will have a received intensity a millionth that of the near-Earth one. This means that the OSS one must transmit at a millionth of the bitrate of the NE one in order to have the same signal/noise margins. So the OSS spacecraft is likely to make GAIA-quality measurements of only a thousand stars, or else lower-quality measurements of many more stars.

That problem could be resolved by using a more powerful transmitter, but beyond a certain point, that will require powering the transmitter with a nuclear reactor.

That said, GAIA will do lots of Nice Things. What Hipparcos did for the Hyades, GAIA will do for more distant star clusters, and also for Cepheid variables -- find distances directly, without going through such intermediate steps as moving-cluster or main-sequence fitting. For more, see this diagram of how distance-measurement techniques overlap (with this parent page).

GAIA will also record the deflection of starlight by the Sun's gravity as it travels into the Solar System; it will provide a test of a prediction of General Relativity with an accuracy of 10^-3, comparable to that of VLBI position measurements.

For more on this sort of test, see The Confrontation between General Relativity and Experiment (rather technical!).