Was the dad bi?

I don't think so, perhaps remotely curious. We see towards the end of the film the dad has a conversation with his son and says "...and I'll say one more thing, it'll clear the air. I may have come close but I never had what you two have. something always held me back or stood in the way."

I don’t think the dad’s confession necessarily means that he had a relationship with a man. It could’ve been a relationship with a woman that was doomed for many reasons. Maybe she was married or Mr. Perlman was already married. Or maybe one of them had to move away and they decided to part ways. I think he just meant to say that he understood Elio’s pain.

I thought he was gay, and it didn't even occur to me until long after I saw the film that this was (apparently) not supposed to be the case.

I thought he was implying that he was gay and (at least) once had the opportunity to pursue a relationship with a man but didn’t.

Plus he loved to study Greek statues of men... I think together with his speech this says that he at least at one time felt drawn to the male sex, or a significant other of the same sex.... and for whatever reason never dared to explore it further...

Yes, he was either bi or gay. After what he said, his son asks him: "does mom know?"

PS. That conversation (or rather monologue) was the only thing I liked about this movie (and maybe Marzia).

There is an interview mit Andre Aciman, where he states that many interpretations of the sentence "Does Mom know?" are valid. He said that he likes it when it stays unclear: https://www.youtube.com/watch?v=h3Fg6aN-7KE (starting at 2:00)