To get around the lack of observed annual trigonometric parallax, Copernicus simply said that the stars were far away (as opposed to saying the Earth was stationary). At the time there was no compelling reason to prefer one explanation over the other. Based on the simple assumptions given above, Copernicus correctly deduced:

• the correct "picture" for the size of the Solar System
• the correct explanation for retrograde motion.
• the explanation for the phases^*, variable brightnesses, and sizes of the planets.

• ....

Despite all of the above successes, the Copernican model was not embraced for sound scientific reasons:

• the model did not explain the observations any better than did the Ptolemaic model; Copernicus was even fored to introduce epicycles because of his assumption of uniform motion on circles
• he simply assumed the stars were distant so as to explain the lack of observed parallax
• there was no physical foundation for the model

^*The Phases of Mercury and Venus weren't observed at the time. Their discovery awaited Galileo and the telescope.