Being that I am a historian and not an astronomer or physicist, I wouldn't know what evidence to produce. Given the history, doubt was removed when Newton developed his laws of planetary motion. Parallaxes work into it, something which was not observed until 1838 (Copernican theory holds that you would observe a shift when viewing a star, though supporters explained the lack of one as being that the stars were too far away to see). It was Newton's work, ultimately, that proved the heliocentric model.
Fact remains he couldn't provide the mathematical, physical, or observational PROOF necessary. He observed compelling evidence ("experimental evidence"), but the evidence he provided couldn't prove he was right--among the evidence, he said the tides were caused by the motion of the Earth (dismissing other evidence to the contrary); he said the orbits are circular, despite Kepler's work.
As for my personal feelings on the topic, obviously the Church was wrong in holding to steadfastly to the (incorrect) interpretation of Scripture. As a Modern Catholic, I can apprecaite that the Bible is a guide, not a end in an of itself. I agree with Galileo (and the modern Church)--the interpretation of the Bible is correct only insofar as it doesn't contradict what is scientifically proven; then, it is not the science that is faulty, but the interpretation. Cardinal Bellarmine said the same thing 400 years ago, but Galileo was unable to sufficiently prove that the interpretation was wrong as shown by science. Newton did that, and the Church should have corrected its position. I'm not sure what else you are looking for...
I don't think Newton proved the solar system. He explained the motion of the planets better than before, but he provided no proof that the earth orbits the sun. And I don't know what parallax has to do with this issue. It was ultimately used to determine the distance to the nearest stars, and the method certainly relies on the solar system model to provide the base of the triangle involved (the diameter of earth's orbit, for observations made six months apart), but that's not proof of the solar system either. Frankly, I don't know of a scientific proof even now. The solar system is a theory, and like other currently accepted theories, it's supported by evidence and it makes useful predictions.
Your insistence on Galileo's lack of proof is not a good argument for your position. He had great evidence, and that's really all that any scientific theory has.