Case is an American maker. The relatives of the Case family had companies making razors in the USA, Kinsfolk, Geneva, Genco...not Parker in Japan.
Parker is very much a Japanese razor. Case and family used steel from Buffalo, New York. Parker and most of the Japanese razors used Swedish steel like everyone else, except for the few Japanese Tamahagane steel razors. The look, grind, finish and blade width are very different as well.