What is the maximum distance measurable with parallax? What is the furthest star or celestial object whose distance has been calculated with parallax and how does it compare to the theoretical limit using today's telescopes? And how exactly does telescope aperture relate to the maximum distance measurable (other than the bigger the aperture the bigger the distance)?
Quick Google reveals a couple simple analyses.  For example, 


  The Andromeda Galaxy, M31, is the nearest major galaxy to the Milky
  Way. The distance to M31 has been measured using other techniques to
  be 
  2.5⋅10^6 light years , or 7.6⋅10^5 parsecs. Using the slightly modified parallax formula, we can find the necessary parallax angle to
  measure the distance to Andromeda. $ p = \frac{1}{d} =
&gt; \frac{1}{7.6*10^5} parsec = 1.3 *10^{-6} arc-seconds $
  
  This is an incredibly small angle. For comparison, the resolution of
  the Hubble Space Telescope is 0.05  arc-seconds, so even Hubble would
  not be able to detect the necessary angular shift of the nearest
  galaxy to effectively use parallax as a measure of its distance.