Heathrow photo
Good photo of Heathrow! It illustrates well the amount of open space to the north of the airport for airport expansion.
It also shows Staines Moor, a "site of special scientific interest" (SSSI)
to the south-west of LHR between the reservoirs.
It has been suggested that a north-south rwy could be built here. Apart from any other considerations, it would only be usable for crosswind take-offs to the south (or crosswind landings from the south), because of conflicts with the existing rwys.
As such it would be of limited use and not a good investment compared to parallel rwy(s) north of the airport.