At familysearch.org you'll find extracted data of death certificates from the State of Washington. The database includes a Nettie Barney who died Feb 22, 1939 and born ca 1853. Her parents were Dwight Burroughs and Elizabeth Chase. Try as I might I couldn't find her parents in any other record. Also in this database is a Geo. W. Barvey who was born Aug 14, 1849 and died Jan 10, 1917. This is no doubt the George W. Barney you are looking for. Unfortunately his parents' names were not listed.
Try as I might I could not find George and Nettie and the two daughters in the 1880 census (and I tried every technique I could think of). A George with a wife Ella was listed in the 1880 census next to Jacob, but since it didn't list any children it makes it doubtful Jacob was your George's father.
I suggest getting George's obituary. It probably doesn't list his parents but it might have the town he was born in and any surviving siblings. Check with a local library or historical society to see if they can help you. I also suggest tracking down the two daughters born in New York. With luck their death certificates, marriage records or obits list exactly where they were born.
Quite often when a person didn't know where his parents were born it's because one or both died when he was young.