Through the late 1940s, the term was definitely a list of songs, not a list of records. Typically, in those times, when a song became a hit, it was recorded by several different artists. In later years, such rerecording was called covering a song, and often rejected by fans of particular artists.
As rock and roll became popular, it was more difficult for generic singers to cover the tunes. It is said that Your Hit Parade was nearly cancelled after many weeks of unsuccessful attempts by big band singer Snooky Lanson to perform Elvis Presley's "Hound Dog". The program finally ended in 1959.
The term is still used, as in the title of the popular magazine, Hit Parader.