Believe Caroline Emily came to NZ with her family sometime before 1901 when she was married to Henry Desmond Geach Grattan
If she married in NZ in 1901 - won't her marriage cert give her parentage?
Have you checked when she arrived in NZ and where she sailed from?
Is Farr a common name all over Wales or is it specific to certain areas.
Just taking the 1901 census as an example
1901 for Wales has 434 entries for the surname Farr of which only 266 were born in Wales
The highest concentration of the surname was in Glamorgan 112 and Monmouthshire 111. Only 36 were in Breconshire
The 1901 for England has 4,726 entries for the surname Farr
The 1901 for Scotland has 93 entries of which only 61 were born in Scotland