Parallel sailing was the navigators inability to determine his longitude. Not a mathematical solution in the sense that the other sailings are, it involved converting the distance sailed along a parallel (departure), as determined by dead reckoning, into longitude. It is the interconversion of departure and difference of longitude when a vessel is proceeding due East or due West. This was a common occurrence when the sailings were first employed several hundred years ago.
Without knowledge of his longitude, the navigator of old found it necessary on an ocean crossing to sail northward or southward to the latitude of his destination, and then to follow that parallel of latitude until the destination was reached, even though this might take him far out of his way. Because of this practice, parallel sailing was an important part of the navigator's store of knowledge. The method was a crude one, and the time of landfall was often in error by a matter of days, and, in some cases even weeks.
The formulas for Parallel Sailings are:
DLo = p sec L
P = DLo cos L