Hi Lynne,
It is great to hear you are conducting some research on your family history.
It would seem possible that "Athany" may have been misspelled for "Athenry" which is a town located approximately 25 kilometers east of Galway city. Roots Ireland has the parish records of
Co. Galway (pre 1864) for this period which may be fruitful. You can also search by first name, surname, and year.
Let me know if you have any other questions and I would be glad to help.
Cathaoir