How is parallax used to measure the distance of the moon?

Parallax is a process of using angles to measure distance objects.  Your eyes are instrument for your brain to use this process to measure how far an object is so that we can see in 3D.

The link below explains more in greater detail.

Parallax is the effect where a nearby object appears to move relative to more distance objects when viewed from two different positions.  You can see parallax by holding your finger up and looking at it with one eye.  Then look at it with the other eye and it will move relative to the wall or sky beyond.

The distance that an object moves (t radians) depends on the distance between the two observing positions (p) and the distance to the object (d).  We usually measure relative to a background which is so far away that we consider it to be fixed.

Strictly the formula for the parallax distance is p/tan(t).  For astronomical objects, this is usually approximated to p/t because the angles are so small that t = tan(t).

Observations of parallax of the moon can be made against the fixed background of stars.  Simultaneous observations are made at different places of, for example, the time that a star is eclipsed by the moon.  Care has to be taken that the two observations are made in line with the lunar motion so that the star is eclipsed at the same point on the limb of the moon.  Or observations can be made of both the entry and exit, then the time to cross the centre of the moon calculated.  We know how fast the moon moves, so the difference in the times of the eclipses can be converted to an angular difference in the position of the moon.

Put your index finger in front of your face centered on your nose, about 1 foot away.

Open and close one eye at a time. Notice that your finger appears to move back and forth. This is parallax.

By measuring the angle between the two "fingers" OR images taken of the sky, astronomers can calculate the distance to the object. This works just fine for objects out to about 2300 light years. After that the angle gets so small that it cannot be accurately measured.

Usually images are taken 6 months apart and the entire "baseline" of the Earth's orbit is used to calculate parallax.

It's a function of basic trigonometry.
http://www.astronomynotes.com/starprop/s2.htm