They are listed in the 1910 through 1930 censuses as living in Baron Co. KS. In the 1900 census they were living in Clarion Co., PA.
The 1880 census of Sugar Creek, Armstrong Co., PA lists William V. Adams 13 as the son of James Adams 50. Also listed are his wife Mary A. 52 and daughters Lula 16 and Rachel 11. The name is misindexed as Adamus.
The 1870 census of same lists James Adams 40, Luly 6, Willy 6 and Rata 3.
The 1860 census of same lists James Adams 30 with Rachael 64 and John 21. Coincidentally they are living next to an Adam Varnum.
The 1850 census of same shows Rachel 54, James 20, Margaret 17, Sam'l 14 and John H. 12.
The 1900 census of same shows Captain James Adams, born March 1830 and his wife Mary, born Sept 1822. They were married 30 years. More than likely his first wife died by 1870 and this Mary is his second wife.
The 1890 veterans census shows Captain James Adams of same had served in Co K, 63rd PA Infantry 1861-1864.
A WWI draft card for a James Frank Adams of Barton Co., KS shows he was born Sept 4, 1891 in Phillipston, PA which coincidentally is in Armstrong County, PA.
The 1880 census of Berwick, Columbia County, PA shows an Allie Phillips 12 as the daughter of A. S. Phillips 49 and his wife Amande 44 (don't assume it is the right Allie, verify, verify, verify).
Some of these records you can look at for free at familysearch.org and Ancestry.com. For those you cannot see for free I suggest checking with your local libraries to see if they have Ancestry or Heritage Quest.
I suggest first getting William and Allie's death certificates and obituaries. The death certificates should have their parents' names as might the obituaries which may also have surviving siblings.
Next get a copy of their marriage license application which would be at the Armstrong County Register of Wills. It may have data on their parents.