I searched the Irish Family History online database at ROOTSIRELAND.IE. It has abt 20M B/M/D records for most Irish counties. Records before 1863 are increasingly sparse: almost non-existant before 1800.
For Mar(..) Friel B 1840 +/- 5 yrs, with a father forenamed Pat(rick), somewhere in Ireland, I found 2 records. Neither were in Donegal (one was in Mayo, the other Armagh).
I couldn't find a marriage record for a Mat(thew) McLaughlin and Mar(...) in 1862 +/- 5 yrs. Are you sure of the names/dates?.