Can anyone explain why from the XVII century standpoint, heliocentrism required stars to be huge, much bigger than Sun? Which observation or measurement hinted at that?

Yes there was lack of observable star parallax. But XVII measurement precision was so, so much worse than the one required to detect parallax (15" was best precision achieved at the time vs <1" required to tentatively detect parallax and ~0.3" required to be sure about it), that stars could "be" 50x closer than they really are, and 50x smaller, and the world would still "fit" the observations if one assumes that parallax is just tad small enough to escape detection. Which means, nearly every star being much smaller than Sun. Such a small star is impossible, but since source of the stellar energy wasn't known anyway, that couldn't be determined.