Take II...
I'm getting confused with mother's names! In 1861, mother's name was Ann, in 1871 mother's name shows as Jane. The daughter Agnes who shows in 1871 age 16 was born in 1855 in Mauchline, parents William W. and Ann McKee (McKie). Her birth cert in 1855 is the magic year for certs in Scotland

The first year of official reg. in Scotland and entries ran over two pages of the register. Her birth cert shows parents as married in 1848 in Grey Abbxy, Co. Down (sorry, I'm useless on Irish place names!). William and Ann show as born there. Their fourth child, 2 boys and 1 girl previously (fitting that 1861 census entry). Ann McKie died in Mauchline in 1861, some days after the 1861 census (parents William McKie and Agnes Harrison).
It would then appear that William Snr. remarried Jane Tait in 1862 in Mauchline, they then go on to have son Andrew in 1865.
So, James's parents (assuming we have the right entry) were William W. and Ann McKie from Co. Down (hopefully you can make out the place name) who married there in 1848. They look to have come over to Scotland (Ayrshire) sometime between 1851-2.
Monica