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Maybe this is them in 1861?

Reference RG09, Piece 1557, Folio 60, Page 6
Baynerds Court, St Clement, Truro

Docking, Francis  Head  M  43  Labourer
Docking, Susan  Daughter  F  14
Docking, Francis  Son  M  10
Docking, Ellen  Daughter  F  4
Docking, Mary  Daughter  F  2              all born Truro

After marriage, the couple are here in 1901:

Reference RG13, Piece 4291, Folio 65, Page 21
Beech Street, West Ardsley, Yorkshire

Asquith, Sam  Head  M  24  Coal Miner Hewer 
Asquith, Mary  Wife  F  23
Asquith, Mary E  Daughter  F  2
Asquith, Annis  Daughter  F  10 months    - all boen West Ardsley


There is a potential death here:

March qtr 1948
Spen Valley district    Vol 2d, Page 439
Asquith, Mary    age 70

But, she may have remarried?

There are 40 James Parkinsons born in Lancashire in 1856/7, but none in Manchester!

Of course, living in Wales he might have said Manchester to make things easy, but could have been born anywhere close by - Salford, Bury, Ashton under Lyne, Chorlton - just to name a few towns!

I feel you absolutely MUST get his marriage certificate, where his father's name at least will be shown. Names of witnesses could also give clues.

Reference for the marriage is:
March qtr 1877
Merthyr Tydfil    Vol 11a, Page 475

1851 census has 2 William Greet's! But the one that matches 1861 is:
Reference HO107, 2089, 475, 1
3 Sanvy Gate, St Margarets, Leicester

Greet, William  Head  M  59  Framework Knitter  b Blaby, Leicestershire
Greet, Jane  Wife  F  52
Greet, Elizabeth  Daughter  F  27  Framework Knitter
Greet, Edward  Son  M  17  Framework Knitter
Greet, Frederic  Son  M  15  Framework Knitter
Greet, George  Son  M  13  Framework Knitter
Greet, Mary  Daughter  F  7
Greet, Emma  Daughter  F  5
Greet, Elizabeth  Granddaughter  F  2         - all born Leicester


and the other:
Reference HO107, 2088, 150, 14
3 Granby Street, Nags Head Square, St Margarets, Leicester

Greet, William  Head  M  60  Framework Knitter  b Leicester
Greet, Sarah  Wife  F  69  Seamstress  b Whitwick, Leicestershire


Finally, the first William in 1841:
Reference HO107, 604, 18/13, 18
Sanvy Gate, Needle Gate, St Margaret, Leicester

Greet, William  M  45
Greet, Richard  M  20
Greet, John  M  20
Greet, William  M  15
Greet, Edward  M  21
Greet, Henry  M  10
Greet, James  M  8
Greet, Frederick  M  6
Greet, George  M  4
Greet, Jane  F  44
Greet, Mary Ann  F  20
Greet, Bricilla  F  15
Greet, Elizabeth  F  15