Why do simultaneous-ops parallel runways need to be so far apart? In the answers to this question about Denver's widely-scattered pinwheel of runways and this question about Schiphol's runway out in the middle of nowhere, it's mentioned that parallel runways are required, by both European and U.S. regulations to be over a kilometre apart in order to be able to have simultaneous departures or simultaneous arrivals on both runways.  In contrast, if one runway is departing an aircraft while a parallel runway is landing an aircraft, the runways need only be a few hundred metres apart.

Why is there such a large runway-separation requirement for simultaneous arrivals or simultaneous departures from parallel runways?  It can't be for aircraft separation, since an aircraft is allowed to land on a runway quite close to a parallel runway that is simultaneously departing an aircraft, despite the possibility of this resulting in a head-on collision if one or both aircraft drift off track, so what, in fact, is the reason for this?
The simple answer seems to be for wake turbulence. This FAA document, ORDER JO 7110.308C, seems to say it is for wake turbulence reasons in para. 9:


  The geometry of the approach, as well as the lateral separation between the two approaches and prevailing local meteorological conditions, provide the wake turbulence avoidance necessary for reduced separation dependent approach operations.


Figure 3 shows the aircraft arriving in a staggered fashion, 1 nautical mile diagonal separation with runways 2500 ft apart (parallel runways separated by less than 2,500 feet, also referred to as, Closely Spaced Parallel Runways (CSPR)).  The text also describes how the lead aircraft follow a lower glide path than the following aircraft such that  the lead's wake turbulence isn't a factor for the following aircraft.