Question

Why do stellar parallax measurements work only with relatively nearby stars?

Answer 1

Because the change in viewing angle is so small for most stars that we cannot resolve it. We can measure only distances out to about 1000 light years.

Explanation:

Even for the very nearest stars the change in angle we see is very small.

Think of an isosceles triangle whose base is a diameter of Earth's orbit, and whose legs go out to the nearest star Proxima Centauri at a distance of 4.24 light years. For simplicity assume Proxima Centauri cooperatively sits perfectly still relative to the Sun, which is not perfectly true. The base is only anout 16.7 light-minutes across the orbit of Earth. So we find that the apex angle, which is the parallax angle, is only 0.769 arc seconds!

We can measure angles down to about 0.003 seconds with ground-based telescopes on Earth (see the reference given below), so we can measure the distance to Proxima Centauri. But the 0.003 second minimum limits the legs of the triangle to about 1000 light years, which misses most stars even in our own galaxy.

For a good discussion of parallax measurements of distances to the stars, see:

http://spiff.rit.edu/classes/phys301/lectures/parallax/parallax.html