Ah, so it was always obvious to you, huh? Okay, longie ....
Anyway, I suspect it would require more than merely measuring the separation between two stars. First, the odds are that neither would be close enough to us to exhibit any parallax. Second, if by chance you happened to observe one that was close enough, you wouldn't know which of the two had "moved." You'd need to get a very precise visual fix a large group, and compare that with a sighting of the same group six months later. Then you could see which one had changed position. That's why I assumed (wrongly it seems) that photography must have been involved.
Bessel used parallax to determine the distance to 61 Cygni announcing his result in 1838. Clearly to succeed it was important to choose a star which was close to the Sun. His method of selecting a star was based on his own data for he chose the star which had the greatest proper motion of all the stars he had studied, correctly deducing that this would mean that the star was nearby. Since 61 Cygni is a relatively dim star it was a bold choice based on his correct understanding of the cause of the proper motions. Bessel, using a Fraunhofer heliometer to make the measurements, announced his value of 0.314" which given the diameter of the Earth's orbit, gave a distance of about 10 light years. The correct value of the parallax of 61 Cygni is 0.292".John Herschel, when he learnt of Bessel's achievement, wrote to him describing it as: "... the greatest and most glorious triumph which practical astronomy has ever witnessed."
Olbers, told of Bessel's achievement on his 80th birthday, said it was a gift that: "... put our ideas about the universe for the first time on a sound basis."
The Royal Astronomical Society awarded him their gold medal to mark this achievement.
Parallax can be measured quite accurately for some distance. A second of star movement (in the degrees/minutes/seconds system) over the measuring time (something like half a year) equals 3.26 light years.