How do I convince my family that being an ice cream store manager is a good career?

Best way to prove them wrong is to find success doing what you want to do. I don’t understand why if you’ve wanted to do this for years you aren’t actively pursuing it by working at an ice cream store. Heck, if you have started working there when you first were interested in it, you probably could have been promoted to a manager position by now. But you’ve wasted years arguing with them about going to college or not instead of doing anything productive at all.

If you want to convince them, then go get the job and show that you can live independently and support yourself as an ice cream store manager.

It's a job, not a career unless you own the store.  It's also not likely to pay very well.  Eventually they will let you go for someone that they don't have to pay as much.

If they weren't convinced after any of the 500 other times you asked this question, I don't think they're ever going to be.  

I think that one can be proud of any job well done.  And there are many different ways to get an education.  However, that said, it is easier to get an education when you are younger and there is plenty of time in life to earn money.  Education is worth way more than just getting you a better job, but it can do that too.  You don't have to follow what your family says automatically but you should at least listen to the people who care about you.