A cat has no concept of a scratching post, i.e. for a cat everything is a scratching post if it is not wet, hard as stone or crying when being scratched. Scratching serves at least three different purposes.
- marking of territory. scratching marks are visible to other cats. Additionally pheromones from the paws are applied at the scratching marks
- stress relief
- sharping of the claws
As there are rather instinctive impulses making a cat scratch, it most likely uses the next best place to scratch if in a certain mood.
What can you do? You have to make sure, that either there is a much better scratching post to make it take a walk to the better post or to put the scratching posts in vincinity to your door.
Most scratching posts are vertical things. When observing outdoor cats you'll realize that they prefer inclined posts, because it is more comfortable for them to scrath them. On a inclined post cats can control their scratching much better by using the weight of their own body. I use two small logs or branches leaning agains the wall. All my cats preferred these leaning branches immediately over all other things.
Best if the new ergonomic scratching post is close to the favorite door.
If that doesn't the trick, I suggest to fix an aluminium liner over the damaged area of the door with double faced tape. This will be disliked by most cats. Keep it there until your cat is used to the other scratching posts.
Using pheromones like Feliway can help bias your cat's preferences.
One last thought. If you close your bedroom door the answer is different. The cat just wants into that room, no matter how. In most cases it is best and simplest to give in and let all doors within the cat's interest open. Many cats can be very obstinate concerning their territory. It makes little sense to resist besides cases where you have to protect an allergic from your cats secretions.