The online parish records for Killedan parish in Mayo go back to 1834 for marriages, but only to 1861 for baptisms, so there won;t be formal birth or baptismal records prior to then (the civil birth records did not start until the 1860's either).
I looked at the page you mentioned, and the marriage on 9 March 1862 was between Martin (Martinus) Higgins and Margaret (Margarita) Carolan. The witnesses are Thomas (probably Tadhg) Carroll and Hugo (probably Aodh) Forde (a common surname in the area - you can see more Forde's on the facing page). The Latinized forms of the names were created by the parish priest, and are not the names which the people actually used, although some of them eventually "stuck".
Interestingly, I think my great-great-grandfather, Tadhg (Thadeus) Gallagher, is shown as a witness on the facing page, at the marriage of one of my other Gallagher relatives.