Yes. That is what the age of consent MEANS - once you're over that age, you can legally consent to sex with anyone else also over that age. So sex with them if you both want to have sex can't be rape. The law says you're old enough to make up your own mind.
What the age of consent does is say that under that age, you cannot consent legally to sex, even if the reality is you have a mind of your own and you want to. So if someone older has sex with you and you're under the age of consent, they are guilty of statutory rape (or "sexual assault of a child" or whatever else your state's law calls it). That is to say, it's not real rape in the sense that you didn't want to have sex but the other person forced themselves on you, it's only rape because the statute law says you're too young to be able to consent.
I know, it's weird, what the law is saying is that you have no brain about sex until one day, bing, you reach that birthday, and suddenly you can make up your own mind about having sex. We just have to draw a line somewhere to protect children from paedophiles and not even the different states can agree about where to draw that line!
What your parents think is an entirely different question. About the most they can do is apply for a restraining order to keep the other person away from you.
And what do you mean by "being with" someone? If you just like to be together and never do anything sexual, maybe you might call that dating, but friends do that too, so making any law about dating would really run into huge problems. I'm going to go out with my mother tomorrow to help her at her bank (I'm an accountant and she likes me along as sidekick so nobody will confuzzle her!), and we'll have lunch and do a bit of shopping. You could date someone and do all that. The only difference is WHY you're with them and how can anyone tell the difference unless they can read minds? So the law has nothing to say about dating, only about sex because that's something we can say did or didn't happen.