How do I get my girlfriend's parents to like me?

Anonymous
So I recently moved out of my home to the state of my online girlfriend. We met in person five years ago, and she visited twice. Her mother offered for me to stay at her home until I find my own place but now regrets it- though she told her daughter, my girlfriend this, and not me. Our time together is heavily restricted, and I'm not allowed to take her on any dates even though we're both adults. I need to earn their trust, and they.. need to accept that their daughter is her own person with a life they can't plan out for her. I can't do anything about the second part, though. They are still helping me, even if it's out of obligation so I owe them. My girlfriend is saying she doesn't care about their opinion. She expresses a desire to run away, 'like she does with everything else,' but I know it will hurt her more in the long run if she just moved in with me. I don't want to hurt her relationship with her family. So, sorry for all the long stuff here.. any advice?
How do I get my girlfriend's parents to like me?
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