A friend of mine is getting married in a few months, and the wedding happens to be the same day as my birthday. I have no problem with this, as I'm happy to celebrate with my friend, and birthdays don't matter that much to me. The problem is that birthdays matter A LOT to my parents, and they have a habit of planning elaborate parties for family members without letting us know in advance. If I tell them I'm busy on my birthday, they'll be insulted. They'll also think I'm being presumptuous for assuming they can rearrange their plans to have a party for me on another day. Do I skip my friend's wedding to keep the peace in my family, or do I find a way to explain to them that I can't celebrate my birthday with them this year?
You don't say how old you are, but since you have a friend who's getting married, I'm going to guess that you're past the age where it's socially expected for your parents to throw you a birthday party every year. It's nice that your family is so close, and it's sweet that your parents continue to want to celebrate your birth with you, but it might be a little too nice, too sweet, too stifling.
For starters, if you want to go to your friend's wedding, then you need to plan to go. Regardless of your age, and whether or not you've ever stood up to your parents before, it's time to start asserting your own identity separate from theirs. Again, there's something sweet about your parents wanting to celebrate with you year after year, but it also sounds like a way of reinforcing for you where you came from and who's responsible for your existence. If this is a pattern just around birthdays, maybe, in a typical year, you roll with it and smile graciously when party-time arrives. If this is a pattern of controlling your social engagements other times of the year, too, then you need to think about what kind of relationship you'd prefer to have with your parents, and if you can help them to see you as an adult with appropriate boundaries.
For this year, you say to them as soon as possible, "Jake is getting married in a couple months. I'm so happy for him and can't wait to celebrate this great occasion. The wedding is on my birthday, so I wanted to let you know now in case you were thinking of planning something." Then leave it at that and see if they get the message. If they protest or say that it simply won't do not to get together, you could say, "I was thinking that maybe this year instead of a party, we could go out for dinner the weekend before." You're giving them the joy of being able to celebrate, but you're sparing yourself the overblown and potentially embarrasing experience of having to endure another parent-thrown party, plus you're setting the terms for how to celebrate.
It's possible that the restriction imposed by your friend's wedding this year will help your parents realize that you can maintain a close relationship regardless of how your birthday goes, and your overall relationship may improve as a result. If they really are insulted, then give them some time to cool off and enjoy the wedding (and your birthday!) free from their expectations. Then decide how you need to approach your parents in a healthy and mature fashion to position yourself for a more practical and independent relationship moving forward.