The (old) long-term car park to the east of the Harrods complex is a candidate for better utilisation, and the area north & east of taxiway alpha is within the airport boundary. If the terrain there was flattened, I'm sure a fair number of additional aircraft stands could be constructed.
Using open surface for car parking is extravagant use of scarce resources - aircraft can't park in multi-storeys but cars can! There is plenty of land available for multi-storey car parking - all along the northern edge of the airport for example. Judging by the extortionate price of car parking, the payback period would be easily manageable.
At present quite a large proportion of the stands are used by corporate aircraft, that could be converted to airline use, with minimal cost and disruption (and likely to win planning approval and obtain financing). South Stands are the primary candidate, then stands 17-19 (the 'Pond'). Adding 8 or so extra stands to passenger use would increase the total available for this purpose by 25%, enough to create some headroom at peak times. The corporate users would need to relocate to the east.