How can GAIA's accuracy for position, parallax, and annual proper motion for a given star be the same number? Per Wikipedia's Gaia (spacecraft); Objectives:


  
  Determine the position, parallax, and annual proper motion of 1 billion stars with an accuracy of about 20 microarcseconds (µas) at 15 mag, and 200 µas at 20 mag.
  


As @RobJeffries painstakingly points out to us in comments below this answer when we talk about parallax as a unit, it is referenced to a baseline of 1 AU. It doesn't refer to the actual cyclic motion seen in the sky due to the orbital motion of whatever is moving around the Sun at its own particular distance.

But the accuracy of both "position" and "annual proper motion" have nothing to do with 1 AU and are measures of positional accuracy of the telescope and how its astrometric data is analyzed.

How can all three of these accuracies simultaneously be 20 µas for 15 magnitude stars?
Parallax and proper motion are determined from a series of position measurements taken over the course of (for Gaia DR2) 22 months. A "5-parameter" astrometric model is fitted to these position measurements, consisting of a sky position at some epoch, a parallax and a proper motion in each of the celestial coordinates.

The precisions of each of these parameters improves as the number of measurements increases, but parallax and mean position precision are basically unaffected by the timespan of the observations (as long as they are taken over more than a year), whereas proper motion precision is improved by a longer timespan given the same number of observations.

If a star had no proper motion, then a basic parallax measurement would be the difference between two (extreme) positions on the sky, and the parallax precision would be around $\sqrt{2}$ of an individual position precision. The mean position of the star is something like the average of the two measurements, so might have a precision that is a factor 2 better. 

In practice, you have many more than 2 observations and measuring the parallax and mean position is like trying to fit a sine wave (with a DC offset), of known period (1 year), where the mean position is the DC offset and the parallax is the amplitude.  In such cases, the mean position has a variance of $\sigma^2/n$, where $\sigma$ is the uncertainty in a single position measurement and $n$ is the number of observations. It took me a while to track it down, but it can be shown that the variance of your best estimate of the amplitude is also $\sigma^2/n$, when the frequency is known and $n$ is large (e.g. https://pdfs.semanticscholar.org/df9b/0c876b488be088fd416387866cf8499fa438.pdf). Thus we expect the mean position and parallax to have similar uncertainties and for these to get smaller as $\sqrt{n}$.

Proper motion precision is different. If we imagine a set of measurements where parallax can be ignored (e.g. at the same time every year), the proper motion precision is akin to the precision in the gradient of a line fitted to two or more points (with time as the x-axis). The precision improves with both the number and precision of the position points and with the timespan of observation.

The answer to your question is therefore coincidence. You have picked on a magnitude of star for which the typical individual position measurement precision, typical number of measurements and typical timespan lead to a rough coincidence in the precision of the 5 parameters when proper motion is expressed in units of per year. For other magnitudes or for stars observed for an atypical number of times or duration, that isn't the case.

Quoting the abstract of Lindegren et al. (2018)


  For the sources with five-parameter astrometric solutions, the median uncertainty in parallax and position at the reference epoch J2015.5 is about 0.04 mas for bright (G &lt; 14 mag) sources, 0.1 mas at G = 17 mag, and 0.7 masat G = 20 mag. In the proper motion components the corresponding uncertainties are 0.05, 0.2, and 1.2 mas yr−1, respectively.