You don't say who the father of Annie J was.
Looking at the parish registers for Boyle, I cannot see a birth for Annie in 1862. It is possible the date is wrong and it might be worth your while looking through the records for other years to try to locate the record. You can access the records through this link. The records go back to 1793 for births and 1792 for marriages with a few gaps, but you may be lucky and find a marriage for her parents there too.
Griffiths Valuation in 1858 has Henry Burnett in Keadue and Harrison Burnett in Cartronaglogh both in Kilronan parish. Again I looked here for the birth of Annie but could not find it. It is probably worth looking through these registers to see if you can glean any more information about the family.
irishgenealogy.ie has a marriage index record for Harrison Burnett in 1848. Unfortunately the record is not viewable at the moment but you can obtain it by following the links on the page.
There is a death record for Harrison in 1896 which includes the name of his son in law which might be helpful for you.
This is a death for a Henry Burnett.
If you can clarify which man is Annie's father I will see what else I can find for you.