The Accuracy of Galileo's Observations and the Early Search for Stellar   Parallax
  The question of annual stellar parallax is usually viewed as having been a "win-win situation" for seventeenth-century astronomers who subscribed to the Copernican view of universe in which the Earth orbits the Sun and the Sun is one of many suns (the fixed stars) scattered throughout space. Detecting parallax would be solid evidence for the Earth's motion, but failure to detect parallax could be explained by the stars lying at great distances. Recent work pertaining to Galileo's observations of double stars illustrates Galileo's skill as an observer. It also indicates that, given the knowledge of optics of the time, Galileo could expect his measurements to be accurate enough that they would have revealed stellar parallax had it existed. Thus parallax was not a "win-win situation" after all. It could be solid evidence against the Earth's motion, evidence which fortunately did not dissuade Galileo from the Copernican view. 
Christopher M. Graney