Why is the Moon’s parallax zero when the Moon is directly overhead?

The premise that the Moon’s parallax is zero when it is directly overhead is correct. In astronomy, parallax is defined as the difference in the direction of a celestial object as viewed from an observer's position on Earth's surface versus the center of the Earth.1

When the Moon is directly overhead, it is at the zenith—the point on the celestial sphere directly above the observer. In this alignment, the observer, the center of the Earth, and the Moon all lie along the same straight line. Because there is no difference in the direction of the Moon as seen from the observer's position and the Earth's center, the angle of parallax is zero.21

Mathematically, this is expressed by the formula:
$$\sin p=\frac{a}{r}\sin z$$
where $p$ is the parallax, $a$ 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 at the zenith, $z=0^{\circ }$, which results in $\sin p=0$, confirming that the parallax is zero.1