If memory serves right
Runwys with 4300 feet+ runway separation are considered as independent runways (no limitations on parallel landings).
Separation between 3400 feet to 4300 feet would require aids like PRM (already mentioned in this thread) for "independent" operations (IMC). [PRM has it's own procedures and 2 additional controllers monitoring a PRM approach with special ground automation support (e.g., MSP)].
Less than 3400 feet separation would require some kind of "dependent" approach in IMC conditions. For example, SFO, with its 750 feet separation is using SIOA/PRM (simultaneous offset instrument approach). Under VMC with visual clearance, you can perform non-SIOA operations, but speeds should be adjusted to not pass the aircraft on approach to the other runway.
Wake vortex considerations are significant below separation of 2500 feet.