How is it safe and legal to land in parallel runways
Whether Parallel Runway Operations are allowed at an airport is dependent on many factors, such as:


Proximity of runways to each other
Availability of precision approach systems for each runway


There is two modes that are interesting here, which is the full parallel mode, where lateral separation is no longer required once both aircraft are established in the precision approach and staggered mode, where the runways are too close for parallel mode and need to maintain 2.5 NM lateral separation during the approach phase.

The same regulations apply to departures, you can read about it in EUROCONTROL's SkyBrary.

Related questions:
How are go-arounds safe with parallel runways?