Looks like the first child was born fairly soon after the marriage: Mary Jane Lusk 26 Aug.1865. IGI also lists James (1867), Agnes (1872), Abby Fleck (1875), James (1877), Charlotte (1880) but you need to check these as at least 1 is a submitted not extracted record.
First complete census for Ireland is 1901 and only small fragments for a few areas of earlier census records survive.
By 'parish records' do you mean church records? To find church records (if the exist) you need to know the family's religion and where they lived. You can check LDS catalogue online (
www.familysearch.org) - do a place search- to see if the church records you need are there and if they cover the dates you want. If so, for a small fee you can order the microfilm from your local LDS library to be viewed there.
If Arthur Lusk came from Scotland you might be able to get census and other records. His father's name and occupation should be on marriage certificate but do you know his mother's name or when Arthur was born?