Where exactly is the historical Korean military demarcation line, the 38th parallel? Where is it shown on the map?

The exact location of the 38th parallel is the border between North and South Korea. It is shown on the map as follows:

According to the Korean Armistice Agreement, the demilitarized zone between North and South Korea on both sides of the 38th parallel is about 4 kilometers wide and 248 kilometers long. It is important to note that the 38th parallel is named for its proximity to the 38th parallel, but it is not the same as the 38th parallel. The city of Kaesong, for example, which is under the de facto control of North Korea, is located south of the 38th parallel.

Expanded:

On September 2, 1945, according to Allied agreement, Supreme Allied Commander MacArthur, in his first directive, used the 38th parallel north as a boundary as a temporary demarcation line for the U.S. and Soviet armies to receive the surrender of the Japanese forces in Korea and to conduct military activities against Japan, respectively, with the area north of it being the surrendered area for the Soviet forces in Korea and the area south of it being the surrendered area for the U.S. forces.

In October, the Soviet Union handed over the administrative power north of the 38th parallel to the North Korean people, and by December all Soviet troops had withdrawn. Six months later, U.S. troops also announced their withdrawal from the south, but left behind civilian officials and a "military advisory corps."

The division of the Korean Peninsula's land and people resulted in the cutting off of transportation, telecommunications, and exchanges of people and goods on both sides of the 38th parallel.

On June 25, 1950, North Korea, with the tacit approval of the Soviet Union, crossed the 38th parallel in an undeclared war to carry out military operations. The Korean War, which lasted three years, broke out and on Oct. 1, 1953, the U.S. and South Korea*** signed the Korea-U.S.*** Same Defense Treaty, under which the U.S. permanently stationed troops in the southern part of the Korean Peninsula.

References:

Baidu Encyclopedia - The 38th Parallel