Triangulation doesn't especially add much (or, in fact anything at all) when the base of the triangle is 2 AU across (the diameter of the Earth's orbit, or roughly 16 light minutes) and the long axis of the triangle is 13.4 billion light years.
I would have thought the differential to movement and the accuracy of a 2AU baseline was meaningful there. And occultation, both of which comes from study of other objects of known distance, which are within the limits of resolution for a 2AU baseline. I believe the GAIA mission pushed to 10-6 arcsec. I can't find a more recent limit online, love to know what it is. (copied from https://astronomy.swin.edu.au/cosmos/t/trigonometric+paralla...)
Given what you know from your interest already you might like to, say, work out at what distance objects appear to essentially fixed in position when viewed from either end of a 2 AU baseline.
Having a background wallpaper of 'fixed' objects that can be occluded helps to inform us about the closer occluding objects .. but triangulation doesn't assist in depth sorting all the distant objects that appear fixed.
My comment above assumed (perhaps incorrectly) that you were referring to the title topic and implying that our motion about the sun was enough to determine the distance to these far objects as the JWT moved through a few orbits.
I think the solar system moves ~48 AU around the galactic centre every year. And the Milky Way is doing ~117 AU per year. (Using google for calculations)
So maybe that combined with space based telescopes helps to make the parallax method more useful over time. Depending on vectors, etc.