I'll let the resident astronomers give the authoritative answer. But:
Given two stars, it seems that the ratio of distances from the observer to each of the stars would bear on the magnitude of the apparent change in relative position of the two stars as the observer displaces. It also seems that the angle of separation from the observer's perspective would bear on it, especially if the observer displaces perpendicular to the bisection of that separation angle.
The worst case would be a very near star which is observed collinear with a very distant star (i.e., obscuring it). As the observer displaces perpendicular to this line of sight, the nearby star would appear to displace reciprocally, revealing the previously hidden distant star. The degree of this apparent reciprocal displacement would be given by quantifying the discussion above, which I'm frankly too lazy to do this afternoon.
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