I have some doubts about those parents for William H. Phillips. I went through the 1860 census records for Monroe Township, looking for anybody named Philips or Phillips, and that 51-year-old William Phillips is one of the records I found and recorded in the Research Notes for William H. Phillips Jr. The census records where William Jr. indicated a birthplace for his father (1800 U.S. census, 1900 U.S. census, 1915 Iowa census, 1920 U.S. census) consistently say that his father was born in Ohio, but the 51-year-old William Phillips recorded in the 1860 census was born in Virginia. Ohio and Virginia are different places, and are not often confused with each other. Considering that the names William and Phillips are both rather common (goodness knows that many men and boys named William Phillips turn up in search results -- and I even found a William H. Phillips, only age 7, in Monroe Township in 1860), I do not think it is reasonable to connect the elder William as father without more evidence than the fact of living in the same town in 1860. If the elder William had said he was born in Ohio, I might think differently.
ADDED: Another census record that Ancestry and FamilySearch have shown me as a match when I searched for this man is a Wm H Phillips, age 17, recorded in 1860 in Greenup, Cumberland County, Illinois. He was born in Indiana to a father named David Phillips who was born in Ohio, so the geography of birth is correct. However, our William H. Phillips enlisted in the Union Army from Indiana in 1861 (I think from Delaware County), which makes it less likely that he lived in Illinois in 1860.
Also, there is a 6-year-old William Phillips, born in Indiana, recorded in the 1850 census in Madison, Jefferson County, Indiana, with father William U Phllips and mother Julia. Those parents were born in Kentucky, which led me to reject them as candidates.