You should be able to find the names by two means:
a pay website with parish and civil ecords from 1864
has civil records:
Examples only as I couldn't find the exact date of Sept 24, 1864
James Gilligan birth registered #594 Tobercurry, Sligo
2 June, 1870 Fa: James Gilligan Mo. Mary Quinn
Anne 22 April, 1872 & Thomas 12 April 1878 siblings
James Gilgan born 28 Sept 1864, Riverstown, Sligo
Fa: Michael Mo: Catharine Davy
James Gilgan born 16 Nov 1870
Fa: James Gilban Mo: Mary Feeney
Look to James's children for clues.
Needs much more research and too many for me to peruse.
There are 13 Gilligan's in the Sligo telephone book, 1 in Riverstown. Some letters might be in order. Address sufficient for mail delivery.
There are two census substitutes for Ireland prior to 1901:www.genealogy.nationalarchives.ie
for Griffith Valuation - Sligo 1858.
There are 62 Gilligan/Gilgan's. These would probably be
James's grandfather's time.
Riverstown or Riversdale was a Village in the Parish of Kilmacallan and also Parish Kilshalvy and Poor Law Union of Boyle, within 12 mi. of Boyle
There were 11 Coen's in Leitrim in 1858 at the time of the Griffith Valuation. Names Patrick, John, Bartholomew,
Martin and James
In 1858 no Gilligan or Gilgan specifically in the Parish of Kilmacallan nor the Parish of Kilshalvy. There is no saying someone moved to those parishes later or had lived their earlier.
Both the spellings appear to be correct as in the Grifith
Valuation in 1858 there were 91 Gilligan's and 60 Gilgan's on www.askaboutireland.ie
but another source just has the name spelled Gilgan (perhaps for convenience)
If need more help don't hesitate to reply by clicking on the reply at the bottom right.