It sounds like this is about you and your ex AND you and your parents. As far as your ex goes, you want to connect with him in spite of all that went wrong, and because it was a long term relationship, I think you should. People make mistakes and he behaved badly, but if this was an isolated incident, or if there was more to the story than you’re telling me, or if there was some part you played in this incident that you may not have realized yet, it will be good for both of you to talk. Relationships overcome infidelity all the time, and it’s rarely just about one person behaving badly — although it could be! So, yes, I think you should talk to him and discuss what happened, why, and if there’s a reason to get back together again or not. Just because he made an impulsive decision doesn’t mean you should, too.
As for your parents, understand that you’re all forging a new part of your adult child/parent relationship. They want you to be safe and happy and to not be hurt, or to put yourself in harm’s way. They’re angry at your boyfriend for hurting you and they see him as the problem. You have the difficult and necessary job of explaining how much you love them for caring, and how much you want them to know that you are going to do your very best to make smart choices in life, but that you need to discuss this transgression with your boyfriend and make your own decisions and that you hope that they will be understanding and supportive because you need them — even when you both disagree.