How do you measure the distance to the stars with parallax?

Simply stated;

you take a picture of the sky (many in fact)

and then take a new set 6 months later.

In six months the Earth will have moved about 185 million miles (base line).

Now compare photos. For some of the near stars you can measure their angular displacement (as compared to the more distant "fixed stars".

You're got angle and base line - that's parallax.