I'm a California lawyer, but to answer your question, no the police do not have to cite you with a crime immediately. In your case, a DUI. It is quite common for the police to not cite you and send the police report to the District Attorney for determination on if a case should or should not be filed against you. As most of you are already aware, the police determine arrest, and recommend charges. It is the prosecutor who decides if charges are actually going to be filed or not. Once the case is filed, then it is sent to the Superior Court, at which point the case finally becomes a criminal matter.
It is quite common in a case like yours for the police to dispense with actually citing and arresting you. What the police did when they found your car is run the registration to see what address and phone number it went to. Hence, the police called your parents. Your parents essentially told him that you had the car last they saw, and so they were able to deduce that you drove the car. They then were able to find out where you were from your parents (your friend's house).
When the police had you blow into the breathalizer, they got the evidence they needed to charge you with DUI. Chances are the District Attorney will file on you. You should be receiving a letter in about 3 months notifying you that charges have been filed, and it will list what court and what date you need to appear. Rolling your car is quite a big deal. You leaving the scene also didn't help your case either. Plus, you are under age. So, the cards are really against you in this situation. Save money for an attorney while you still can. You have 3 months to save up. You will need one.
Jail may be in your future. Really consider going to AA. That will at least show the Court you learned your lesson and won't make bad decisions like drinking, rolling your car, and leaving the scene. You are lucky you didn't die or get someone else hurt. I wish you luck! And try not to do anything that has a good chance of killing you.