was the priest their biological father?

The big twist was that the priest was in cahoots with the crazy twin sister.

But why was he so attached to the sisters in the first place? It's possible that he was their biological father, but had to hide it to keep his occupation since Roman priests have to be celibate. The duplicity perhaps drove him crazy.

Why was he on crazy Mary's side rather than Julia's side? Because Julia's dream at the beginning of the movie suggests that she was the one who permanently disfigured Mary's face while she was sleeping to get back at her for being an oppressive "master" (remember, Julia later described her relationship with Mary as that of slave and master respectively). Julia obviously blocked out that she did this and so, years later when she went to visit Mary at the mental institution, Father James (their literal father) lied to Julia about the reason why Mary's face is disfigured.