How is separation managed when there are only two parallel runways in a very busy airport?
With two runways, usually, one is dedicated to arrivals, and the other is dedicated to departures. That's the simplest way to handle separation -- only in-trail separation is involved. 

That's usually the way traffic is managed at Heathrow (LHR) and Dubai (DXB), two of the world's busiest two-parallel-runway airports. At Heathrow, when in a westward flow (70% of the time) the runway functions are swapped about 3pm, to equalize noise impacts.

San Francisco (SFO) has two parallels, 28L and 28R, that are 750 ft apart -- very close. Under visual conditions, both runways are used for VISUAL approaches. Generally, aircraft arrive side-by-side, for wake avoidance.

For about an hour a day. 6am to 7am, when arrivals dominate, LHR allows arrivals on both runways. I don't know what separation rules are used. At SFO, under marginal conditions, both 28s are used for arrivals but aircraft are staggered and capacity is less than for visual operations.