As a general suggestion, you should give some dates for Mark/Michael and Sarah, and information on their children if they are no longer living. The same for the Hugills and Parkers - full names and dates.
I did find:
Mark Foley, d. 1924 aged 53, in Barrow-in-Furness district.
Doreen Foley, b. 1914 in Ulverstone district (similar area), mother's maiden name Teasdale.
http://www.1911census.co.uk/ - in Ulverstone there do appear to be a family of Foleys:
Mark Foley, 43
Sarah, 38
William, 11
Louisa, 7
John, 4
Ellen, 2
at
http://www.1901censusonline.com/ they are indexed in Dalton in Furness in 1901 as "Forly":
Mark, 31, b. Wales Rumney, Labourer Shipyard
Sarah, 28, b. Durham Stockton
Patrick, 5, b. Yorks Granton
Mary J, 3, b. Yorks Middlesbro
William, 1, b. Lanc Dalton
The marriage is indexed at freebmd.rootsweb.com:
Dec 1894 quarter, Middlesbrough district, Vol 9d page 1181
Sarah Teasdale and Michael Fooley.
I think I have found him in the 1871 and 1881 census - in 1871 in Cardiff with his parents (who are both indeed from Ireland) and in 1881 with parents and younger siblings in Liverpool, but it might be best to leave that until his father's name can be confirmed from the marriage certificate.
If the child who went to the US is the son William, he is on
www.ellisisland.org, and that gives you an address for his father in Dalton in 1922, and the name of an uncle, William Taylor, already in the US.