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Here is the short answer to ‘what is parallax’: it is an apparent displacement(difference) in the apparent position of an object viewed along two different lines of sight. It is measured by the angle or semi-angle of inclination between those two lines. Objects that are closer to the solar system have a larger parallax than those that are far away, so parallax can be used to estimate celestial distances.
When looking at the distances from star to star(stellar parallax), parallax created by the different orbital positions of the Earth causes nearby stars to appear to move in direct relation to more distant stars. If you observe parallax, measure angles, and apply geometry, it is possible to determine the distance to various objects. Stellar parallax is most often measured using annual parallax, which is the difference in position of a star as seen from the Earth and Sun. Annual parallax is normally measured by observing the position of a star at different times of the year as the Earth moves through its orbit. Measurement of annual parallax was the first reliable way to determine the distances to the closest stars. Stellar parallax remains the standard for calibrating other measurement methods. Accurate calculations of distance based on stellar parallax require a measurement of the distance from the Earth to the Sun, now based on radar reflection off the surfaces of planets.
Parallax can be figured out because it is a special form of the triangulation principle that says you can solve for all the sides and angles in a network of triangles if the length of at least one side has been measured and all of the angles in the network are known. So, with the careful measurement of the length of one baseline you can fix the scale of an entire triangulation network. In parallax, the triangle is extremely long and narrow, and by measuring both its shortest side(motion of the observer) and the small top angle (always less than 1 arcsecond and other two close to 90 degrees), the length of the long sides can be determined.
The true importance of parallax may never be realized. With out modern system of propulsion it would help to decide fuel loads, but, since we could never travel those distances on a fossil fule, it may never come into play.
We’ve done many episodes of Astronomy Cast about Parallax. Listen here, Question Show: Distance in Space, Changing Earth’s Orbit and Different Sized Stars