Yes there are minimum distances to be respected but I don't have the numbers.
I am sure of this because that's the main problem in DXB where they have 2 huge parallel runways and even staggered BUT not enough to be used as independent parallel runways.
So they have to respect some separation between landing aircraft on one runway and aircraft taking off from the parallel one...
You should find your answer somewhere in the ICAO document where the criteria for parallel runway operation must be specified.