Your logic is flawed.
As long as it's your Parent's house all the doors in the house are Theirs too!
And,,in effect, even the ' Room' they currently allow you the privilege of occupying is also Their's.
And all the more effectively, since it's Their house, Their rooms therein and Their doors therein & throughout it's also Their rules!!
Witch means they can lock the their own bedroom door like a bank vault and literally burn every other door in their house & replace them with shower curtains if they want to.
In fact,, there is no law that requires them to have any doors or curtains in their house at all.
Neither is their any law that says they can't provide you with clothing from the Second-Hand store,
Or, requires them to Feed you any more than a Thousand calories a day even if it's just Dog-food as long as it's not labeled unfit for human-consumption!
So I suggest compromising with your parents.
Replacement doors and non-lockable knob-sets are available at Low's & Home Depot.
Otherwise, Pet Smart has some tasty alternatives and you might even find a cute little dog-house with a with a curtain on the entrance that you can use in the back yard,
Since as long as it's also Their Property, technically the Law doesn't pacifically require them to provide you with any more 'Shelter' than that either or as much as a lawn sprinkler on the end of the water-hose for bathing!