Four years later I answer my own question. Arnold's Quality Comics were actually published by "Comic Magazines" of Buffalo, New York. So Parent's Press could have started a "Quality Comics, Inc." without stepping on his toes. They didn't put "Quality" on the covers. I don't know why they bothered spinning off a subsidiary when they were already publishing comics under their own name. Probably some kind of tax dodge.
Link to the book:
Calling All Kids 08