[PatrickHenry]

I think (RA will correct me if I'm wrong) that the lack of any visible parallax shifts wasn't something that could be dealt with until the development of photography, which made possible very detailed records of star positions. The photos could be compared, when taken six months apart, to finally detect the minute apparent movement of the very few stars that are close enough to exhibit a parallax shift. Then, knowing the size of earth's orbit, and with a little high-school trig, their distances could finally be determined. I think that was one of mankind's greatest intellectual accomplishments. With better telescopes, I think there are now hundreds of such stars.

[AntiGuv]

I'm unsure exactly when and how stellar parallaxes were first observed, but my recollection is that it was not until the 19th century. That would've been very close to the invention of early photography so that may very well have been the prerequisite. Hmm..

[dread78645]

I think (RA will correct me if I'm wrong) that the lack of any visible parallax shifts wasn't something that could be dealt with until the development of photography, which made possible very detailed records of star positions.

After some Googling it seems Friedrich Bessel made the first parallax measurment two years before John Draper made the first daguerreotype image of the Moon.
Modern measurment are done with photography and spectroscopy, but it seems the pioneering work was done with sextants.