Where I live, which is on the edge of the Peak park (but not in it, maybe a couple of miles away) and an area of special beauty, the local council are really difficult when you want to put stabling or field shelters up. I have been turned down for stables and they are not happy about field shelters either.
My goats and llama have an open fronted shelter we built ourselves, and waited with bated breath for complaints, but go away with it. I have been in discussion with the council about a field shelter and they are not very helpful at all with saying yes or no. I get the feeling that perhaps there is nothing they can do if its open fronted and no door, but they are not saying so. I know if its mobile and you can prove it moves - if its on skids, the council cannot stop you.
I know that horses are domestic pets in the eyes of the council. If you can fit stables in your garden, you do not need planning permission.