The role of John The Baptist of the New Testament echoes/recalls that of Elijah the Prophet of the Old Testament.
echo (v.)
To repeat details that are similar to, and make you think of, something else:
The design of the church echoes that of St. Paul's Cathedral.
Cambridge
recall (v.)
To cause you to think of a particular event, situation, or style:
His paintings recall the style of Picasso.
Cambridge
If you are writing, I would avoid the informal group genitive: "[John The Baptist of the New Testament]'s role" because it can be (mis)read as the role of the New Testament. (We get away with this construction in speech because we start a sentence and paint ourselves in a corner and the group genitive saves us from rephrasing. But when writing we should rewrite "The boy I saw on the corner's father is my teacher.")