Did the father know?

I'd say he definitely knew. During his conversation with Thomas at the bistro, he says he left the house because it didn't make sense for him to live there without their mother however, that's proceeded by him saying it was also time for the boys to live by themselves. This implies he knew, because why else would they need to be alone? Why not sell the house and buy a new one for them all to live in if neither son had left home yet? Also, he is caught in the middle of a video chat between them where Francisco upsets Thomas by implying he's been playing around and seeing other people and the dad realises it's inappropriate for him to be listening, yet Thomas insists he stays online and asks Francisco outright in front of his dad if there is a reason for him to be jealous. Lastly, when Francisco and the dad meet up for lunch, the dad asks him if he wants to take about "this" - i.e. the reason he's been going out so much and having so many late nights. Francisco quickly cuts him off as he doesn't want to go into it, but establishes it's because he misses Thomas so much. The combination of those three scenes definitely points to the dad knowing, however just like his wife, the mother, he allows them to continue their relationship without interfering.

I too am a little disappointed this wasn't explored more, at least a conversation between the dad and another adult such as Rosa about the boys, however having read about the film, it was a deliberate decision by the director to portray as pure a love story as possible and not about the outside world's view of it. I appreciate the love story and was utterly convinced by it however you can't stop yourself from wondering how it would work in the real world.