Why is the Moon’s parallax greatest when the Moon is near the horizon?

The Moon’s parallax is defined as the difference in the direction of the Moon as seen by an observer on Earth's surface compared to an observer at Earth's center.1

This parallax is greatest when the Moon is near the horizon because the observer's position on the surface is at its maximum angular distance from the zenith relative to the Earth's center. Mathematically, the parallax $p$ can be determined using the formula $\sin p=\frac{a}{r}\sin z$, where $a$ is the Earth's radius, $r$ is the distance to the Moon, and $z$ is the angular distance from the zenith. When the Moon is directly overhead ($z=0^{\circ }$), the parallax is zero; when the Moon is on the horizon ($z={90}^{\circ }$), $\sin z$ reaches its maximum value of 1, resulting in the greatest possible parallax, known as the horizontal parallax.1