Biographical information on Parker can be found here and here. David Apatoff deals with a recent book about Parker here.
Today, he is not nearly as well known to the general public as Norman Rockwell. But that could be said as well for successful contemporaries such as Coby Whitmore, Jon Whitcomb and Edwin Georgi whose work appeared in many of the same slick magazines as Parker's. Beside being very good at what he did, Rockwell's fame is based on the fact that he painted cover illustrations for the Saturday Evening Post, America's leading general-interest magazine in its day, and those illustrations were self-contained stories. On the other hand, Whitmore, Whitcomb, Georgi and Parker mostly illustrated fiction pieces in magazines, the illustrations themselves often evoking the story subject, but not in themselves being self-contained visual narratives.
Worse for Parker from an historical standpoint was his strongest professional attribute, an ability to change his style, sometimes in the form of creating new illustration style fashions. This is in contrast to some other illustrators who had strong, easily-recognized styles that provided fame and fortune ... until fashions changed and they wound up having trouble getting work. Parker's career was long and successful, but it can be difficult to immediately identify many of his illustrations without looking for his signature.
There is one major exception to the previous statement. Below are examples from his long-running series of mother-daughter matching outfit covers for Ladies' Home Journal, the leading women's magazine in American for many years.
These images are the largest I could find for those covers. They are included because they clearly demonstrate Parker's ability to alter his style.