Common at major US airports (eg LAX) is two pairs of close runways either side of the terminals, four in total. Normal that on each pair, as one touches down, generally on the outside, a takeoff starts on the other. Just as the OP described, and similar spacing. Over on the other side of the airport the same thing is going on, quite independently.
The old Chicago O'Hare configuration, until about 10 years ago, was fascinating. 6 runways, in a sort-of Star of David pattern, intersecting, terminals in the middle. They used to use them all at once, both for takeoffs and landings, all very impressively interleaved with one another.