I don’t think my bf trusts me

So tonight while my bf and I were on the phone he mentioned about a past issue that we had before. Long story short he told his friends parents who he used to live with that I cheated on him a few months back. Which I didn’t I absolutely love my bf and I would never cheat on him I’m just not that person. So what happened was I was driving a male coworker home who I happen to be good friends with. We ended up talking in my car for an hour or two and when my bf called to see if I was ok I didn’t pick up and told texted him I would call back in a bit. When I eventually called him back he asked me what I was doing and I told him what was up and ofc he was upset. I understand his hurt, I should have picked up the phone first of all and secondly I was alone in the car with another man who he hasn’t met before. But all we did was talk about work, but my bf is convinced that I cheated on him. He says that he trusts that I didn’t but he has no proof and it seems sketchy to him and that it will always be this way. Now back to today he told his friends mom that that happened and that he also hasn’t met my parents, which I haven’t really met his dad except for a brief moment, and we’ve been dating almost a year. I’ve asked him to meet my parents but my dad wasn’t interested my mom met him pretty much in passing. I just feel like he won’t like my family personally. But he told his friends mom that and I was kind of hurt and shocked that he would say those things about me and then state that he didn’t paint me in a bad light and that he did believe me that I didn’t cheat. Idk I see that he is wanting more in our relationship and I am happy about that but I’m also hurt that this scar is on our relationship and I don’t know how to prove to him that I didn’t cheat and I never would. PLEASE HELP!!!