Well, there is the fact that independent parallel arrival is not currently a procedure approved for LHR by the CAA
Chicken and egg.
There has been no reason up to now to develop the safety case for independent parallel arrivals, given that runway alternation has been enshrined in government policy for the last 40-odd years.
But with the watered-down alternation scheme that will result if a third runway is built, simultaneous landings on adjacent runways would happen around 50% of the time and the CAA have already published their initial assessment of SOIR at LHR.