Andrew Ernest Young son of the people shown was born on 29 December 1885 (birth registered 1886). The parents' address does indeed seem to be Ligfordram, which appears to be in the union of Newtonstewart in co Tyrone. His sister Matilda Jane was born on 26 June 1887.
Small points worth noting are that Andrew's birth was reported by his father who made a mark (although the marriage register gives the impression that he had signed). Matilda Jane's birth was reported by Mary Jane Greer who was present at the birth (perhaps Matilda Greer's mother or some other relative), who also made her mark. The marriage was witnessed by Matthew Greer, probably a relative. A Matthew Greer Labourer giving the same address married in 1870. His father was another Matthew Greer, farmer.
There is absolutely no point whatever in paying for a service to search for births, marriages and deaths for this period. The underlying registers are available free on Irishgenealogy.ie courtesy of the Irish government. The search there is not absolutely the best; if you don't find what you want try searching FamilySearch first and then look at Irishgenealogy.ie when you have the details of where/when an event was registered.
There are lots of other Young births in Strabane during the 20 years after the Young/Greer marriage, which you could check.
Unfortunately, the civil BMD records do not go back far enough to help you much with the previous generation; you need to look for church records (on the same site, or FamilySearch), or wills & other documents eg here.
