"Parson Brown" is the term used to talk about a typical angelican priest of the eighteenth and nineteenth centuries. "Parson Brown" is not an actual person (though he might have been at some time), but a figure of speech, like "John Doe" is an unidentified male and "Charley" is a watchman. "Parson" by itself means minister.
The classic line from Winter Wonderland:
"In the meadow we can build a snowman
Then pretend that he is Parson Brown
He'll say 'Are you married ?', We'll say 'No man,
For you can do the job when you're in town.'"
Sure, it sounds like some kind of sexual reference, but its not. It's just a priest asking a couple if they are married. They reply that they are not and that they will wait for him to "do the job" of marrying them when he comes to their town.