More context would help, as it is a lot of this is guesswork.
L could stand for a person or a place, normally someone or somewhere that had been mentioned previously beginning with L.
A parish is the smallest level of government of an area with its own church, in the UK anyway.
In the UK Cumberland was a county that no longer exists, it joined with Westmorland to become the new county of Cumbria.
A county is considerably larger than a parish so Cumberland parish must be the American one in Virginia.
A person can’t be a parish, nor can they be a county or town council.
His son is not a parish.
Increase refers to future generations, not just children.