The fact you are trying to do your best on your own is admirable. Sometimes you just need some help. When you try as you have tried, your parents are more than glad to help some. Accept that help in those times with true appreciation. We cannot do everything on our own, especially when we are younger. It is the mooch who continually asks for help without first trying that earns the ire of those around them. Your honest attempt is what your parents will find admirable. They love you and want you to succeed. Sometimes that means a controlled fail, sometimes that means helping you out. It sounds like you are trying. Don't be disheartened that they learn of your struggles. It means you are growing and a regular peraon.

Congratulations! You've shown your dad that you actually are independent - not only did you have the foresight to save up for a rainy day, you didn't need to ask him for help when you had an unexpected expense. Yes it sucks but that's life - and you were prepared for it.

You're doing a good job showing them you're independent. It sucks that it's taking your savings to fix, but you were independent enough to HAVE those savings. And you got the ride covered without asking them for help. Car break downs are a part of life and they know that. You handled it on your own. If that isn't independent, what is?

Welcome to being a new adult. This happens. And yes, 'independence' is kind of a myth--as you've just learned, people rely on each other a lot for things, and that's actually a really healthy part of life. Your boyfriend is giving you a ride to work now; you'll give him a Netflix password later, that's how these things work. People taking care of each other is what we're here for.