How to fix thing with my gf and her parents?

 I really like my gf we have been going out for 4 months now.shes 18 and im 17 a few months ago my gf mom saw a picture on facebook on me and my gf and she was sitting on my lap and kissing me. After she saw the picture she said we was not aloud to date or see each other anymore. So since its the summer time and we are out of school and we are still not aloud to see each other I went to her house 2 days ago with my moms car witch I know was stupid on my part. I only went over the for 5 minutes we just messed around but we did not have sex.Yesterday me and my gf were texting about what we did that day that I was over her house. She had left her phone on her bed to go pick up something in her room. Then her mom comes in her room and my gf dives on her bed for her phone then her mom instantly knew something was up and read all the messages. Yesterday was my birthday and her mom knew that. After she read the messages she instantly called my parents. My parents were really pissed and yes I know what I did was bad and I take full responsibility for my actions I just want to find out a way to gain my parents trust back and find a way to fix things with my gf and her mom. My gf mom treats my gf like a baby we cant date over a picture she would not even let her see me on my birthday before all of this happen but what really got me is that her mom told my parents what happened on my birthday. I'm a really nice guy I just make really stupid desestions sometimes. Can somebody help me please I really love my gf and want to be with her. I tryed apologizing to her mom before about some stuff in the past and she still thinks we should be friends and not date i need help. My parents dont want me to have contact with any of them period but I just dont like it when I know that someone does not like me I just want to fix the problem I was gona write my gf mom a letter and apologize for my actions and ask if we can meet up and talk face to face but Idk if thats a good idea please help

As I am a mother with a daughter of teen yrs.. I have to put it to you the way we think.. When a parent see's there child with some one as you said the picture.. It's a common reaction for her mother might not see her as grown up yet and is trying to protect her... But the mom needs to come to realize her daughter has grown to be a young lady.. I don't think the picture was bad and you had done anything wrong on your birthday.. But if you have the feelings for this girl as you say you do. You need to write that letter and I would suggest hand it to her in person. Ask her to hear you out.. Tell her that you hadn't went against her wishes until your birthday.. She might understand. Teen parents are very common these days. So you need to make sure that you can't honestly tell her your true intentions with her daughter.. I'm not sure if you guys have done anything other then the reagurlar affection.. But I'm sure she is afraid your too young for a miss hap and that's why she has reacted this way. The only way to tell is find out her feelings and what her fears are. Go see her. Try to make it when your gf isn't home.. That away she doesn't seem like you guys are ganging up on her and make the situation worse.. I hope I helped. Good luck. :0)