My bf doesn’t trust me

So let me start from the beginning. A few months ago I was talking a coworker home from work and me and him are good friends, so we end up talking for a hour or two in the car just about work and things like that. My bf calls and looking back now I immediately regret ignoring the call and texting him that I would call him back later. This hurt him a lot and honestly I really regret it idk what I was thinking. But this ended up hurting him badly. He feels that I cheated on him and that I shouldn’t have been alone in the car with another man that he hasn’t met before for a few hours. Especially since I declined the call. We ended up arguing about it and he was mad for a few days but eventually he said he believes that I wouldn’t cheat but it still seems sketchy. Which is totally understandable. But I don’t know how to prove to him that I didn’t cheat. So that brings us back to last night when we were on the phone he said he went over to his friends family house who he hadn’t seen in a while and they asked about me. So he told them that I could have cheated on him and that he still hasn’t met my parents. So the parents thing since we started dating has been weird. I’ve met his dad once only in passing, and we had plans to get dinner with him but from my memory his dad canceled but he said that I didn’t want to go. I remember being nervous to meet him because I just have anxiety when I meet new people but I knew it was an important step so I wouldn’t cancel that. He met my mom in passing when he dropped me off at my house once. I wanted him to meet my parents but my dad doesn’t really have any interest in meeting him because my bf because he is black. Plus idk if he even would really like my family. And in past relationships I’ve always kept them away from my family because I get anxiety that the person I’m dating won’t want to date me anymore after meeting them. It’s just from a past relationship that I’ve had criticized my family when we were breaking up saying that they were weird and stuff. To me it sounds like he wants to take more serious steps in our relationship but he feels that I might not be ready or he’s pointing these things out to convince himself that I’m not worth it or maybe that I need to fix these things before we take that step idk. I just don’t know what to do to convince him that I didn’t cheat. I know in the back of this mind it will always be there hurting him and that hurts me. The parents thing I plan on fixing as soon as I can. But yeah he told his friends family all that and I was kind of shocked and hurt when he said that and then said “I didn’t paint you in a bad light”, and “I don’t think that mom likes you and they are shocked that we haven’t broken up”. I just don’t know what to do. PLEASE I NEED HELP!!