My parents have forgotten my birthday twice;
the first time was when I turned 8 years old, and everyone was just really busy with stuff. I didn't really care, though. Birthdays have never been very hyped up in my family. But when they all realized it about a week later, they bought me a huge cream cake and took me to the zoo as a sort of apology-party.
The second time was when I turned 19. My dad was on a business trip, and my mom was on jury duty. And, to be fair, I also forgot my birthday that day, because I had morning classes, followed by work, followed by night classes, so I was just too exhausted to think about it.
None of us came to realization about that for another 2 weeks, and at that point it was just funny.
I can see how it would be painful for you, though... however, understand that your dad is only a person. Just like you, he gets stressed, he gets busy, and he sometimes forgets things, even important things. He's not perfect, he sometimes messes up, and sometimes you have to be more mature than you would normally need to be and accept his problems and forgive him
If I were you, I would just drop a subtle hint or something, rather than straight-up telling him, because then he'd just feel really guilty.
For example, maybe say "Dad, what was today's date again?" or say "Dad, am I old enough to get a license?" or some sort of question like that, and he'll most likely come to the realization on his own.
If they don't remember... maybe just let it go until next year.
It sounds really difficult, but sometimes you need to buck up and be the calm, collected sane one in the family when everything else is screwy and crazy. And if it's any consolation, it sounds like you at least have good friends who remember your birthday, so that's something.