Parson is another term for clergyman or pastor. So Parson Brown is Pastor Brown. They are telling “Pastor Brown” that they are not married but he can marry them when he is in town.
“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.’”