Why is one of two parallel runways sometimes closed in foggy weather?
SFO makes use of simultaneous close parallel approach operations.  This allows higher density operations than would otherwise be possible for runways spaced so close, but to make use of these ops, in SFOs case, requires 1600 ft ceilings and 4 sm visibility due to the requirement that aircraft maintain visual contact during close parallel operations.

It does look like SFO can run the ILS approaches simultaneously outside of close parallel operations, but is possible the required spacing makes its pointless to use both runways.  If weather was bad enough that cat II ops were in effect, I don't see any verbiage on the cat II charts indicating simultaneous ops are approved.

Regardless of how precise your instruments are, there are strict requirements on how close you can be to another airplane in the approach phase and that separation be maintained if both parallel aircraft perform the missed approach.

The ability to run parallel approaches in low visibility depends on how far apart the runways are laterally.  Airports like IAH and ATL can run triple parallel ops and if you take a look at their runway layout you will find they are spaced quite far apart and may be displaced relative to one another.