The local council planning department will not know. There is no automatic link between planning and land ownership. Though the council tax know, that's no help if the land is not rated for council tax or business rates, and why should they tell you anyway.
Chris` suggestion is the way to go though my previous experience at finding registered land this way is a bit hit or miss. It's fine if there is a postcode, but I have missed finding parcels of land when they are part of a bigger parcel.
For a larger fee than the online method, one used to be able to make an application to LR by describing what the enquiry is adjacent to, and their researchers do it. Not sure if this service still exists.