I found this note on the Web:

{Quote}
According to the liner notes to "Best of Broadside," "Plastic
Jesus" is a parody of an African-American camp meeting song with the line
"I don't care if it rains or freezes, leaning on the arms of my Jesus."

It was the theme song of a religious radio program in Baton Rouge in the 1940s.

"Plastic Jesus" was written by Ed Rush and George Cromarty and was
published, first in Sing Out!, then in Broadside, in 1964.
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