there could be a couple of problems using that method to search, as there is a possibility of a false match due to an incomplete search
1. Not all counties are included in IFHF database (e.g. Offaly, Cork City, Wexford etc etc)
2. Not all available parish records have been transcribed by the IFHF
3. Some parish records for this timeframe have not survived
Cunningham is quite a common surname and was widespread throughout Ireland - top 10 counties based on an analysis of Griffiths valuation :
1. Down 360 *
2. Galway 163
3. Cork 154
4. Antrim 129
5. Roscommon 129
6. Tyrone 111
7. Louth 109
8. Armagh 105
9. Monaghan 95
10. Westmeath 93
(* number of households)
that said with two names with matching father's name, it's probably worth following your Tipperary line of investigation a little further, but I think further evidence would be needed to be certain you have the correct people..
Shane