Hold a thumb in front of you and look at it with one eye at a time. It appears to move against the background because each eye views it from a different position. This apparent shift is parallax.
Astronomers use the same geometrical idea to measure nearby stellar distances. Earth’s position changes as it travels around the Sun, providing separated viewpoints. With the baseline known, a smaller angular shift generally indicates a more distant star. The star need not physically jump sideways to create that apparent movement.