At what minimum distance the direction at which eyes point becomes parallel?
Mathematically, the eyes are parallel when they fixate at, and only at, infinity. Now, because of noise, there is a distance after which it's not really going to make a difference (otherwise you would be able to tell the distance between stars at night). There are 2 parts for your question. First, at what distance it doesn't matter because the eyes won't stay still enough to accurately maintain this vergence angle. Second, at what distance it perceptually makes no difference because you can't tell if the object is closer or not. Concretely it means looking at the distance where each of these metrics is smaller than detection threshold (the value below which your performance on a detection task is lower than a specified value). Conveniently, Chopin et al. (2016) measured both values (these numbers can of course be found in older papers too).

For the first question, they measured vergence noise of 225 arcsec (or 225*pi/(60*60*180)=0.0011radians). The distance for which vergence becomes smaller than that is approximately 55 meters (vergence~=IPD/FD, with IPD interpupilary distance of approx 6cm). So farther than that you wouldn't be able top tell whether they fixate at infinity or not.

For the second question they measured relative disparity thresholds. Relative disparity is the difference in vergence angles between 2 points. So for relative disparity angles lower than threshold you cannot tell whether point A is closer or farther than point B. They unfortunately don't report the exact value but from the graph it looks like 300arcsec, corresponding to a distance of about 40 meters. In other words, after this distance you can't tell whether an object is farther than where you look at.

Finally, you could also ask: at what distance can't I tell whether an object is at infinity or not? The distance of a point in isolation is called absolute disparity, and is known to be much worse than relative disparity (i.e. it is much easier to compare the distance of 2 points than to report the distance of a single point in isolation). This distance turn out to be quite shorter. From the same paper the absolute disparity threshold is about 3000arsec (again, they unfortunately don't report the exact value). This gives a fixation distance after which you cannot tell whether you fixate at infinity or not of about 4 meters.

This is all very approximate and making lots of assumptions but I hope this more or less answer your question.

Chopin, A., Levi, D., Knill, D., &amp; Bavelier, D. (2016). The absolute disparity anomaly and the mechanism of relative disparities. Journal of vision, 16(8), 2-2.