Looking for information on Michael McCarthy & his parents - Waterford County

Hi I am looking for information on my Great Great GrandfatheMichael McCarthy & his parents Michael McCarthy & Ellen Power

Michael McCARTHY AKA McArthy (sometimes I find they leave out other C)

Great-great-grandfather1833 - Jan 11 1875

One certificate states Waterford county and another states Waterford City (I don't quite understand this??) He married Mary Hill in Ballarat Victoria Australia (where I am from) They met out in Australia. Michael immigrated

Age 19-20 - Melbourne, Vic, Australia July 18 1853

His parents Michael McCarthy & Ellen Power married
Youghal, Cork, Ireland Nov 4 1827 Apart from this information I can't seem to find anything else I don't think they immigrated as no records of this . I can't seem to pinpoint where these 2 were even born?? I am planning another trip to Ireland as I have just returned and would like to visit and see where they are from. Any assistance would be very much appreciated. Thank you in advance. Michelle