#Show them that privacy isn't about secrecy
The fundamental misunderstanding here is that the reason you want privacy, is because you want secrecy, "if you have nothing to hide, you have nothing to fear" is the famous exploitation of this fallacy, the idea that everything should be public by default, simply because privacy allows secrecy is essentially an excuse used by powers that be to ignore boundaries and restrict freedoms.
Your parents need to understand that having your door shut is, as well as a symbol of trust between you, a practical and useful tool outside of it's use to conceal. Having your door shut, as well as being a simple fire safety precaution, makes it easier to concentrate without distractions from your work, it allows you to make your room into an ambient space, watching videos and playing music out loud and the like, you can do many things in a room with the door closed that you can't do as easily without.
Start here if you don't care about context
So my advice is simply to shut the door and do something that's clearly not secret, but requires concentration or begets privacy, make something, paint something, do a jigsaw puzzle, whatever, and have music playing, and when they open the door, greet them and ask them to shut the door behind them as you're trying to concentrate or don't want to bother anyone with your music. Be civil, what you're doing here is making it seem like a courtesy that you've shut your door. And if they are simply blind to this kind of communication, then your parents have a problem with paranoia that isn't going to be fixed by you right now. This is not an issue worth ruining a relationship for.