This thread sidesteps your attempt at ignoring time in the equation, and, for the sake of argument allows the mileage to the star to be correct. I could debate that, and win, but better to keep it simple here. Parallax is only good out to, they say, some 326 light years if I recall, correct? So, put that in miles and we can allow it for the thread. The thing I brought to light here is that unless time existed at the star, the light year could not represent the time light takes to get from the star, rather only (for the sake or this thread) the miles that we determine from parallax.
Now if the base line for the trig measurement is IN our solar system and OUR space time, then it has built into it OUR time!!! Even the distances could be skewed, but since that doesn't matter, let's avoid getting bogged down on that bit.