I think it is to do with (a) orbital speed and (b) telemetry and (c) power.
In order to measure Parallax you need to measure the position of the star from different locations in the solar system. The Parallax becomes more precise the greater the separation between those positions.
At Earth-Sun L2 you get a difference of about 2 au in 6 months. i.e. the spacecraft has a baseline that changes at 4 au/yr. In a 5-year mission, you essentially get 10 samples of the full baseline, that enables you to beat down the errors by $\sqrt{10}$, equivalent to an effective baseline of 6.3 au. At the same time, because the spacecraft has executed complete orbits, all of the sky has been sampled with a similar baseline (imagine viewing a line tracing out the spacecraft orbit from a distance - it will have a similar length when viewed from any direction).
If you calculate how long it takes a satellite in orbit at Neptune to make a baseline (defined by the chord of a circular orbit) of 6.3 au, it is only 5.5 years.
However, that would only be for part of the sky - the part at right angles to the spacecraft motion. Large portions of the sky would have barely any baseline at all because the spacecraft motion is essentially straight towards it. Solving for Parallax and proper motion (the relative tangential velocity of the stars) would also be difficult if the proper motion was then parallel to the satellite motion. At Earth-Sun L2, this problem goes away because every 6 months the Parallax motion reverses, but the proper motion doesn't. Around Neptune you would have to wait 84 years for that to happen.
Of course you would also get the observational baseline between where the spacecraft journey began (the Earth) and Neptune, which is potentially 30 au. However, this doesn't solve the problem of all-sky coverage and also wouldn't solve the issues discussed below.
The other issues are practical and I suppose potentially soluble if you throw enough money at them.
Gaia has a limited telemetry bandwidth. At the moment there is significant autonomous decision making and processing before a subset of the data is sent back to Earth. These problems become many orders of magnitude harder when you are 30 au, rather than at the Earth-Sun L2 point which is a mere 1.5 million km away.
Gaia also needs power and it uses solar panels. You get about 900 times less power per unit area at Neptune, meaning 900 times bigger solar panels or some alternative (nuclear) power source.
Lastly, it's much harder/more costly to send the same spacecraft to Neptune rather than the L2 point.