The surviving Lamplugh parish register starts in 1581 - and John Dickinson's burial is one of the first to be recorded. His widow Annas [Agnes] remarried eighteen months later - a marriage of nine years - and died some 15 years after that. The spellings Annas and Agnes are interchangable at this time, and presumably indicate that the pronunciation was towards 'Annas'.
From the Lamplugh Parish Register
Burial
Marriage
Burial
1581 Dec 18
1583 June 24
1592 July 2
1607/8 Feb 17
John Dickinson of Streetgat
Leonard Stanwix and Annas Dickeson the wife of John Dickeson of Streetgate
Lennerd Stanwixe of Streatgat
Agnes Stannix relict of Leonardi Stannix
John and Annas had six children - John, Agnes, Elizabeth, Richard, William and Janet. Annas went on to have a further son, Thomas, by her second husband Leonard Stanwix.
The children of John and Annas are listed (usefully, as their baptisms would have happened before the extant Lamplugh register) in the probate inventory of Leonard Stanwix (20 July 1592):
to keep six children of John Dickinson deceased for their children portiones (whose names are, John Dickeson, Rychard Dickeson, Wyllm Dickeson, Janet Dickeson, Annas Dickeson and Elizabeth Dickeson £23 18 2
As John younger married in 1595, it would seem likely that John senior was born around 1540 (both Johns probably married in the normal age range for a yeoman male of 25-35 years old; and there is no evidence to suggest complications).
And that is probably about as much as one can infer about him. He left no will.