Your Dad being a preacher man obviously believed his interpretation of written book. I have met similar over the years, many different bibles in English, one was the Old King James version translated directly from Greek apparently.
'Birthmark' on listening to the song I would interpret as our flaws which each has. Many songs lyrics are written then people attempt to relate to what the meaning is, this song they have not yet found. Although the Transcript is available next to song on YouTube.
Perhaps one day you will understand knowing the situation, perhaps better than most.