I admit to not being up to speed on this issue, but it seems to me all that would be needed to do parallax measurements would be an adjustable micrometer eyepiece and a telescope of sufficient resolution (large enough apeture and good enough optical quality).
Measure the angular separation between the desired two stars, wait several months to create a large enough baseline, and measure again. Repeat on successive years to be sure the change in separation isn't due to proper motion, then compute the distance trigonometrically.
What I don't know is when micrometer eyepiece was invented, or what it's limits of accuracy were, but in theory at least, parallax could be measured without the need for photographic imaging.
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.