Notice in their claim for "Docksum" (Liancourt Rocks) that they wrote that the Korean name meant "pot island" and that it was given that name because it looked like a pot. Apparently, the Patriotic Old Men never saw Liancourt Rocks because they look nothing like a pot. The name "Dokdo" (獨島) actually means "Lonely Island."
The Patriotic Old Men seem to have thought Liancourt Rocks was a reference to either Ulleungdo or Ulleungdo's neighboring island of Jukdo (竹島), which is just two kilometers off Ulleungdo's east shore. Not only does Ulleungdo's neighboring island of Jukdo look more like a "pot" than Liancourt Rocks, the name "Jukdo" (竹島) uses the same Chinese characters as the Japanese name for Liancourt Rocks, "Takeshima" (竹島). The fact that the same Chinese characters were used for both islands is probably the source for the misunderstanding. The Patriotic Old Men must have thought the Japanese were trying to claim Ulleungdo and its neighboring of Jukdo.
As for their claim to Parangdo, the Patriotic Old Men seemed to think they were a "series of islands," instead of an underwater reef. An October 22, 1947 article in the Korean newspaper Dong-a Ilbo HERE talks about a "cluster of islands" called "Parangseo" (坡浪嶼); nevertheless, no one could find the islands, so Korea ended up dropping its demand they be included in the Peace Treaty. It was not until 1973 that a Korean man named Han Gwang-seop (韓光燮) found the islands "for the first time." (See HERE.)