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What does horizontal parallax reveal about the distance to the Moon?

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Diagram illustrating how astronomers calculate the distance to a star using parallax: Earth is shown in two positions on opposite sides of its orbit around the Sun, with lines extending to a nearby star X and further to distant background stars. The apparent shift in star X's position, measured as angle A, is used to determine its distance, with the baseline labeled as 186,000,000 miles. — Parallax Calculating stellar distances.

Horizontal parallax is defined as the maximum geocentric parallax observed when a celestial body, such as the Moon, is positioned on the horizon.12

It reveals the distance to the Moon by representing the angle subtended at the Moon by the Earth's equatorial radius. Mathematically, if $a$ is the Earth's radius and $r$ is the distance to the Moon, the horizontal parallax $p$ is determined by the relationship $\sin p = a / r$ . Because the Earth's radius is a known value, measuring this angle allows astronomers to calculate the distance to the Moon with high precision.13

The modern accepted value for the lunar horizontal parallax is approximately 57′02.6″, which corresponds to a mean distance of about 384,400 km (238,900 miles). While historical methods relied on geometric observations from different locations on Earth, contemporary measurements often utilize radar and laser ranging for greater accuracy.3