There was a recent article in Scientific American about this (unfortunately, subscription required).

Given the size of the image of a star seen thru a telescope, either the size or the distance can be calculated from the other by elementary geometry. Because of diffraction from the wave nature of light -- which wasn't understood back then -- these calculations said that if the stars were the same size as the sun they couldn't be much farther away than the farthest planets. Which would be close enough to see annual parallax if the earth really did move around the sun.

And so because of this and a couple other reasons, Copernicus had to resort to proof-by-theology since the available evidence appeared to disagree with his position.