How do I ensure that the first meeting between my parents and my fiances parents goes smoothly?

Despite their differences, they all love you in their own ways, you're the middle ground on which they meet, the link that ties them all together. I'm sure you love your man and feel comfortable around him, show that to your parents as its in their duty to make sure you end up with a man that takes care of you and with whom you're happy. As for your fiance, he'll be more nervous than you since he's the one taking away their beloved daughter, so you have to be careful not to come across as daddy's/mommy's little girl, its a big turn off and might even scare your man away. You're a grown woman, you have to show your man that you love and respect your parents yet is mature and adult enough to make your own decisions and have your own life. good luck!