Black begins with 1 and 3 which White answers with 2 and 4. The B and W positions run parallel to each other on opposite sides of the board. Note that if 3 were at 4 and 4 at 3 the result would be the same.
When W answers 1 with a play in an adjacent corner, B has the opportunity to play in the diagonally opposite corner. If W then plays in the last remaining empty corner, the two players have created a diagonal or tasuki fuseki.
The famous Shusaku Fuseki is neither a parallel nor a tasuki fuseki in the sense that the players do not start out by splitting the corners between them.
The parallel fuseki that have a high profile are the Chinese-style formations, and sanrensei. These however are complex openings, which branch into very many variations from about move 10. Nirensei has been a major presence in professional games for the past two decades, increasingly adopted by White.
One parallel formation that receives relatively little attention (in the English-language literature) is the Orthodox fuseki.