Generally speaking, no. Installing solar panels is considered to be permitted development (as long as the installation doesn’t extend beyond certain limits). However, for domestic properties, you will need planning permission if “in the case of land within a conservation area or which is a World Heritage Site, the solar PV or solar thermal equipment would be installed on a wall which fronts a highway”, or if “the solar PV or solar thermal equipment would be installed on a building within the curtilage of the dwellinghouse or block of flats if the dwellinghouse or block of flats is a listed building”.
For non-domestic properties you will need planning permission if “in the case of a building on article 2(3) land (i.e. land within a conservation area or which is a World Heritage Site), the solar PV or solar thermal equipment would be installed on a roof slope which fronts a highway”, or if “the solar PV or solar thermal equipment would be installed on a listed building or on a building within the curtilage of a listed building”.
See here for more details. See class A for domestic and class J for non-domestic.
There has been a recent government consultation about these rules, so keep an eye out for updates.
There are numerous conservation areas in Cumbria and the Lake District National Park is a World Heritage Site. If you are in doubt, please check the planning advice page on your local planning authority’s website – some also provide a phone number you can ring for advice. Cumbria County Council provides a summary of the contacts and weblinks for each authority.