Though the parallax method for working out distances has been known since Euclid, even the closest stars are so distant that their shift could not be measured with the necessary precision until the 19th century. The largest stellar parallax, that of Alpha Centauri, is approximately 77 arcseconds. That is less than the width of a finger held at arm's length, and ancient equipment lacked the sensitivity to observe the shift.
Though a shift of 77 arcseconds was, in principle, detectable by the equipment available to the ancient Greeks, Persians and Chinese, astronomers in these societies would have had to know to look at Alpha Centauri in particular, as none of the thousands of stars in the sky demonstrate as large a shift. The number of stars whose shift can be observed without modern observation techniques is so small relative to the total number of stars visible on a clear night that something like a whole-sky inventory would have been needed to find the closest, and easiest to observe, stars.
This inability to demonstrate the apparent motion of stars was used as an argument against the Copernican model of a heliocentric solar system. Tycho Brahe, among others, argued that the lack of an observable stellar parallax suggested an impossibly large gap between Saturn, then the farthest object known, and the nearest stars.