Let's take both the parents and the boyfriend out of the equation for a minute. This choice is not about either of them. Its about you, what you want to do, where you think you'd be happier. Do you feel that if your parents hadn't begged you'd have no problem moving out? If so, don't let them guilt trip you into changing your mind and making you miserable in the process. This is about you, not them.
Now, let's break this down and bring in the opposing forces...
How old are you? Seems irrelvant, but kids move out and you parents need to face the fact that if you are old enough, moving out is just something that's going to happen. Its not to slight them or because you hate them, its simply part of growing up. They can't honestly expect you to stay living under their roof when you're 45 and married with a family, can they? Assure them that this choice you are facing is not because you hate them, and that they won't be cut out of your life, but that you just feel its time to move out and, well, grow up.
Which it seems you need a healthy does of if you're allowing your mind to be swayed by the wants of your parents. At some point, all kids move out. They're not bad or evil for doing so, its just the first step on the path to becoming an adult. I moved out and when with my ex fiance when I was 20 and my parents didn't want me to leave, asked me not to in fact, but when I said my mind was made up, they respected the choice. They knew I had to in order to grow in life. Your parents throwing themselves to the ground in a fit of tears is kind of....overboard....to a normal mile stone in any person's life. And you need to address that issue with them. You need to let them know that this grave over reaction is tearing you apart and making you miserable. That you love them, cause they are your parents, but that you can love them even when living in an apartment across town. It doesn't mean you won't see them anymore and if makes them feel better go all "Gilmore Girls" and set up one night a week that you (and your boyfriend) go to their house and have dinner and agree that all special holidays you will be there for. But they can't prevent you from growing up, that's not fair to you.
With your boyfriend, I can understand his hesitation. They aren't his parents so its not to easy for him to forgive and forget when they go kneebound. He has seen the bad side of them for your whole relationship and with out the emotional family connection, its no wonder he has the bad idea that its a trick. You need to be a little understanding that he's not apt to be as "Oh mommy, daddy, I forgive you!" as you are and he's bound to be skeptical by a complete 180 in attitude.
Let him know how much this whole thing confuses you and not to pass judgement. Ask him to give you a bit of time to work things out in your head, and if he loves you he will respect that.
My advice on the whole thing though...I do kind of get the sense that your parents are playing a bit of a game here. Maybe not in the way your boyfriend thinks, but in some way. They traded one type of restraint for another. If you are old enough to be moving out, then I would consider it. You can get a job, start helping pay bills, start becoming an adult and your parents have to understand that's just part of a person's life. You deserve to be happy, not imprisoned by guilt.
Again though, this all depends on how old you are and what you have under you. Running off as a child with nothing to support you, like a diploma or whatnot, would be a damn foolish idea and your boyfriend would be a damn fool to try and force you into this.