Think about what you are saying. If the background stars are at their claimed distances, then in 6 months or even 5 years, their velocity through space would matter not at all compared to their position in the sky in relation to us and would have no effect on parallax. The further an object is, the less it’s apparent position in the sky would change. Only objects closer than believed to be would be affected by their velocity in relation to us.
Also have you ever considered stellar aberration? Relativity claims that the only velocity that matters is the velocity between source and observer. But stellar aberration does not take this into account at all. Only the velocity of the earth matters in reality.
Stellar Aberration and Einstein's Relativity
Both fall off linearly with distance. See? Both apparent transverse motion and parallax. They both get smaller at same rate as distance increases.
This stuff is only geometry. It's got no complexity in the trigonometry.
(Next regarding the effect of aberration, in contrast to the non-orbital induced aberration, parallax has a 12 month period, and so the component of aberration due to our sun's velocity will have no periodicity (and also should affect all visual field nearby stars about the same). But as best I understand, annual aberration (the component due to orbital motion of the Earth) is
consistent and known and so can be subtracted out.
A couple of things to consider about aberration is that it's independent of distance, and so we'd expect that the reference visual star field all around the target star also has essentially identical aberration all the time, always, as the target star. That's key.
Because of this general situation, I'd expect aberration is easily separated from parallax ( and also we can keep in mind a sufficient number of observations are needed separately just to separate out the actual transverse motion of the target star from the parallax.) In other words, aberration isn't much of a complication.
But about the article link you gave, this sentence seemed well...miskaten in concepts to me: "Observations show clearly that, in contradiction with special relativity, stellar aberration does not depend on the relative motion between the source and the detector but exists only when the detector is moving" >
We already know only the detector motion should matter (by theory). As best I know, light should travel independent of the emitter flowing a 'straight line' (so to speak) of the curvilinear spacetime curvature, and this is why
only the detector motion relative to that spacetime reference frame matters and by theory the emitter velocity does not matter, or so I think (but feel free to check on that!).)
Good news about measuring distance to clusters just showed up. First you may need quick review
How ' Negative Parallax' works in my wording
Here's a news article just showing up about the exciting new and more accurate parallax measurement on a cluster.
https://phys.org/news/2018-04-hubble-precise-distance-ancient-globular.html
I think though we are getting away from the thread topic, but you could always start a thread on some question (or put it in the Astronomy News thread) and PM me.