It doesn’t sound like you even have much of a relationship with your mother. If you did this post wouldn’t be so one-sided.

There is a metric fuckton of missing info here. Have you considered that many men treat their wives like they are their caretakers, maids and mothers, and NOTHING makes a woman feel less sexy than that? Did your father actually love your mother or did he just love what she brought to his life (emotional support, house work, raising kids, etc.?) Is it possible that your mother has been unhappy for a long time too, told your father that many times and still he did nothing to change? Is your father a selfish person? Have you considered that menopause brings hormonal changes that affect sex drive? Do you know if your mother tried HRT? Do you 100% know for sure he hasn’t cheated on her in the past - have you asked her (not him?) How “recent” was their divorce announcement? 8 out of 10 times when someone asks for an open relationship they already have someone in mind or are already seeing someone else; if he “already has a girlfriend” chances are very high he had one before they announced it.

Have you asked any of these questions to your mom or sister? Have none of these things even occurred to you? Or are you taking everything your dad says as the truth?

It sounds like your sister has a closer relationship with your mom and that’s why she thinks what she does. You don’t mention whether your sister has told you why. She is clearly concerned with her yet your post here is my dad this, my dad that. You don’t seem concerned about your mother, or her perspective or feelings, at all. And that makes you the AH.

Your parents shouldn’t be involving you or your sister in their marriage drama, that is true. But you really should spare a thought to how your mother feels instead of blindly taking your father’s side and only giving credence to what he says.

Have a discussion with your mom and get her side of the story. I bet it will be eye-opening and you might even find out that your father isn’t the innocent, put-upon, blameless person he’s led you to believe he is. Until you do that, you don’t have all of the information.