Tolkien seems to back up the existence of a Balrog’s wings in the book at a further point from their first mention.

First mention of possible wings:

“The shadow about it reached out like two vast wings…”

This could be just a simile, or it could mean actual wings. I think Tolkien meant for the second option to be true, as a later passage verifies:

“…and suddenly it drew itself up to a great height, and its wings were spread from wall to wall…”