HST images of the path of Proxima Centauri against background stars. The proper motion is just the constant linear trend with respect to the background stars. It is (half) the amplitude of the curved motion in the picture below that corresponds to the parallax. But the two components can clearly be seen and separated (see below). In the case of Proxima Centauri the motion against the background stars due to proper motion is larger than the parallax. This means you have to do more than two measurements to separate out this component of motion on the sky. Now in reality, it is a bit more difficult than this because stars also have a "proper motion" across the sky due to their motion in our Galaxy relative to the Sun. So 1pc is 1 AU/$\tan (\theta) = 3.08\times10^$ arcseconds (depending on target brightness) for about a billion stars. The "parallax angle" is actually half this angular displacement, and a star is said to be 1 parsec away if the parallax angle is 1 second of arc. Trigonometry then tells you what the distance is as a multiple of the distance from the Earth to the Sun. This angle forms part of a large triangle, with a base that is equal to the diameter of the Earth's orbit around the Sun. You then calculate the angle that the star has moved against its background stars. You do this twice, separated by 6 months. You measure the position of a star in a field of stars that are (presumably) much further way. The currently accepted answer is not relevant for finding the distance to a star like Proxima Centauri.
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