What is the minimum distance allowed between two parallel runways?
In the United States, the FAA has published Advisory Circular 150/5300-13A, Airport Design , which includes standards and recommendations for airport design, including parallel runway separation.

In summary,


"For simultaneous landings and takeoffs using VFR, the minimum
separation between centerlines of parallel runways is 700 feet (213
m)."
For simultaneous IFR operations, "Dual simultaneous precision
instrument approaches are normally approved on parallel runway
centerline separation of 4,300 feet (1311 m). On a case-by-case
basis, the FAA will consider proposals utilizing separations down to
a minimum of 3,000 feet (914 m) where a 4,300 foot (1311 m)
separation is impractical. This reduction of separation requires
special high update radar, monitoring equipment, etc." Note:
Simultaneous Offset Instrument Approaches allow as little as 750 ft
between runway centerlines.


International regulations do differ. ICAO have their own standards. Here is an example of a document which prescribes airport standards compliant to ICAO Annex 14.