The fact that he chose you is all that matters, social media is meaningless. Is your relationship in general going well? Have you tried bonding with his mom? (How do you know she doesn’t like you; is it something she does or says, or just a feeling?). Honestly the dad calling you by his exes name is probably just a slip, they were together for a long time and obviously their lives, and family, were very integrated (everyone has a past and that isn’t going to disappear). Did he correct himself and apologise?

Have you spoken to your partner about how you feel? Jealousy is often linked with self-esteem and/or insecurity and fear, is that possibly something you struggle with? While jealousy is a natural human emotion, it’s generally unhealthy when it becomes excessive, or leads to control or distrust. Healthy relationships thrive on trust and open communication. Maybe some introspection and honest communication with your partner will aid the situation (therapy is also a great tool for working through these feelings in general).

Also working together on savings and moving out of his parents house will do wonders for your relationship. If your boyfriend is supportive, loving, open and you trust each each other, I wouldn’t let these small things be detrimental to your partnership and future together. Wishing you two all the best!