In the song Winter Wonderland who do you pretend the snowman is?

The answer you seek is "Parson Brown"

A parson being a minister and that is why the next lines go on to ask about marriage and that he can do the job.
In the meadow we can build a snowman,
and pretend that he is Parson Brown.

He'll say "Are you married?" We'll say "No man!"
But you can do the job when we're in town!