Pastor living in parsonage yet owns home in another state

by Ben
(NJ)

As a new pastor fresh out of seminary, this is all confusing to me. If I am living in the parsonage provided by the church and own a home in another state which I rent out, do I claim housing allowance for my rental home and also receive free housing via parsonage?

I would assume that is double dipping. The rental income would add to my salary come tax time. The parsonage should not show up as income on my W2 if I am understanding all of the things I am reading but the home that I own will since it would be rented. Thanks in advance for any advice offered.