I think you may have it!
Someone has just told me "There's only one deer park in that area of South Glos as far as I know, that is at Dyrham."
I guess it could be either. Badminton definitely has a Roman settlement and villa in close proximity (as per my EH link), as well as a deer park. Also Badminton is nearer to a standing stone than Dyrham as far as I can tell.