I know that we cannot measure cosmological distances with parallax, but this is more a philosophical question. Am I correct in assuming that if we could measure distances using parallax (like stellar parlax used for nearby stars) the distance we would measure would correspond to the "angular diameter distance" ? https://en.wikipedia.org/wiki/Angular_diameter_distance
Yes, exactly. The angular diameter distance is defined such that it is compatible with "regular" trigonometric rules, just as the luminosity distance is compatible with regular relations between flux, distance, and luminosity.
Only for at flat and non-expanding universe would these distances coincide with each other, and with the "true" distance, which is what you would measure if you lay out meter sticks (after freezing the Universe in time, so it doesn't expand more before you're done).