Communicate
Slamming the door, hiding in your room or refusing to speak to your parent is not going to solve anything- it will actually make things worse.  Rather than offering any kind of intelligent argument, you are only showing your parent your childish and immature side.  By ignoring your parent and shutting him/her out you may be seriously hurting his/her feelings.  Eventually you will have to speak to your parent again; the longer you spend being stubborn and refusing to work things out, the harder it is going to be to make that first move.     

Choose Your Battles
Any relationship is going to start to wear if the only communication happening is angry.  Your first reaction may always be to get mad or defiant, but take a moment and decide if it's worth it.  If you fight about everything, your parents won't take your anger or frustration seriously.  Before you stop talking to your parents for a week again, think over whether or not it is really worth fighting over whose turn it is to take out the trash- a 30 second chore.

Make Compromises
Be willing to make compromises - Keep in mind that your parents are only trying to do what they think is best for you.  By finding the middle ground you are not only allowing them some input into your decision, but you are also showing them that you are able to be mature and think through your choices.  

Don't hold a grudge - Once each of you have apologized and worked out a solution to the problem, it's important to put the incident into the past.  When future conflicts come up, don't dredge up the old fight.   

Say You're Sorry - Sorry can be one of the hardest things to say, but it is also one of the most powerful words we have.  Even if you do not feel you started the fight, you can express the fact that you're sorry it even turned into a fight, or you're sorry for the hurtful behavior that the fight caused you to display.  All the burden of an apology is not on you- hopefully your parent will reciprocate with a sincere "I'm sorry" of his/her own.

Make Good Decisions
It may sound cheesy, but the best way to help your parents become comfortable with your growing independence is to keep on making smart decisions.  Remember, most fights with parents arise because they are worried about protecting you, not because they want to make your life miserable.  The more they see you being responsible, keeping yourself safe and succeeding, the more they are going to trust you to make your own choices.