Hi Al
Not sure whether you have looked at the censuses for some background on William and Agnes. These look like the most obvious entries:
1851:
William Broomfield 41, Drystone Dike Bailder, b. Eccles
Agnes Broomfield 34, b. Coldstream, Berwshire
Adam Broomfield 11, b. Hutton, Berwshire
James Broomfield 1, b .Hutton, Berwshire
Broomfield under 1 Month, b. Chirnside, Berwshire
Town: Chirside, Chirnshire, Berwickshire
1861:
William Broomfield 52, Drystone Dyker
Agnes Broomfield 44
Margaret Broomfield 19
James Broomfield 11
John Broomfield 8, b. Chirnside, Berwickshire
William Broomfield 5, b. Bunkle, Berwickshire
Address: Old Ladykirk, Ladykirk
If William and Agnes died in Scotland, their death certificates should include full parents' details, including mothers' names, if known to the informant. All BMDs are available to view on line at
www.scotlandspeople.gov.uk, pay to view but very reasonable. The birth entry that you have for William son shows in 1855, the first year of official reg. in Scotland. It should include, for that year only, where parents were born, ages, date and place of marriage, how many children they'd had (boy/girl, alive/deceased). Would certainly be worthwhile you looking at to confirm some background on them.
Monica