Is parallax a "technique to measure the star distances" or a "tiny shift in star's position"?
It is both - a small shift of the position of a star on the sky as we see it, and a means of estimating the distance to the star.

The apparent position (with respect to very distant objects like quasars) changes because our viewing point changes as the Earth moves around the Sun in its orbit. The amount by which the position changes is inversely proportional to is distance.