Dear John,
Thank you for having read the essay and for your comment.
Yes indeed the Copernicus model was less accurate than the Ptolemaic one and it could be refuted on various empirical grounds (like absence of parallax). That is the reason why we said in 1st paragraph on page 3 that "despite having no empirical evidence" Copernicus still insisted his view was correct; and many followed him in the following centuries.
As for the parallax, as far as my understanding is concerned, it was certainly considered as a silver bullet by people opposed to a Copernican view but Copernicus himself had an argument to explain the absence of observed parallax with respect to the background of the fixed stars (which can be found in his book On the Revolutions of the Celestial Spheres) which was that the stars of the background of the fixed stars were so distant that a parallax was simply not observable. The absence of parallax was then used to infer the size of the universe. That's what I would call turning the tables!
As you may know this lead to further major problems with regards to the size of each individual star within the background of the fixed stars. Namely, a single star was then expected to be larger than the Solar System. This compelled Tycho Brahe to propose an alternative geo-hello-centric view in which such problems would not occur.
In any case, since as you say the parallax was only observed in 1830, the absence of parallax did not prevent many scholars, including Newton, to adopt a heliocentric view. The way it was done was precisely by overhauling the way to infer the laws of Nature.
Now, if your comment implied that the presence/absence of parallax constituted an absolute observational means to assert who is moving, I would disagree. For the parallax argument to make sense, one still needs to assume that the background of the fixed stars is actually fixed. If this assumption is lifted and this background has no special property then nothing can be derived from observing a displacement of one of the stars.
Best,
Fabien