You are still their baby, and moving in with your boyfriend is a big step toward independence. That is a scary thought for most parents. If you have a child, this will be just as scary for you even though you did it yourself.
It's important not to spring too much on your parents at any one time. This may seem obvious, but introduce your boyfriend to your parents and try to hang around enough so that they will get to know him better. They need to get accustomed to whoever you bring home.
You parents want to know you are safe and happy, so help them to understand why you love this man and what makes you trust him. They need to trust him, too.
One of the best ways to get your parents on your side is honesty. Sounds crazy alright, but if you talk sincerely with your parents about how you feel, you may come to a much faster and better understanding.
The relationship between your parents and you is about to change, and keeping them involved in your life is one way to make that transition easier. Let them help with the task of moving in. Simple things are big in this process, like letting them help you find your new apartment or having their help in decorating the new place. Whatever can bring them in to your decision and not hold them at arms length.
Listen to what they are telling you. They have seen a lot, and even if you have made your mind up, do them the courtesy of really listening. Maybe they will try to talk you out of it, but you might be able to benefit from mistakes they have already experienced. Let them know you consider their advice an asset, and it will help you keep the kind of relationship with your parents that you need while your life is changing.
Your parents want to support you. Help them with that. Make them a part of your decision, and keep them in your future life. It may be awkward at first, but all new things take some time to get used to. Let them know you appreciate all they have done and that you have learned how to survive in the world thanks to them.
This is a very exciting time, and your parents want to experience part of that thrill with you. That way, they can be there if you ever need them, which is all they probably want anyway.