I want to move out of my house as soon as possible. Not because I'm abused or anything. Just because.
To literally answer your question, "When can I leave my house legally with my parents consent?"
Any time. I'm assuming you mean WITHOUT your parents consent. 18 years old. The age of majority in Louisiana is 18.
The answer given here should be considered general in nature and should not be considered legal advice, or that there has been an attorney-client relationship established. It is always suggested to retain the counsel of a lawyer for legal advice specific to your situation.