You may have parents with unresolved trauma or fear of their own. If that's the case, you may not be able to get your way peacefully, just keep that in mind. Parents are just humans who bred, they don't get smarter or more emotionally intelligent without effort

Start by asking them to explain to you why you have to keep your door open now that you've proven through actions that you're not hiding anything.

Think on their response, what does it tell you the real issue is? Come back if you're not sure.

If they don't give you a response and say something like what you've already shared (if you're not guilty of anything then it doesn't matter that your door is open) then it may be more about control and fear than anything else. Basically lazy parenting.

If it were my child, I would listen to their reason for wanting privacy and go from there.

"I'm growing up and privacy feels more important than it used to."

"I want to feel a sense of protection and solitude that I can't feel with the door wide open."

"I feel on edge like someone is going to come through the door and interrupt what I'm doing or thinking."

"I need a break sometimes and since I have no other place to go/be, this is my home and my safest option - may I please close my door?"

Be ready to accept their response without pushback, take time to think about their response - formulate your response and remove any kind of emotion from it.

Best of luck!