Death certificate is available online. Indicates he was born April 16, 1830 in Ireland, and his father's name was John. Mother's name is unknown.Whom did he marry and when? The marriage record may provide the names of the parents. 1900 census indicates he immigrated in 1848 and his wife in 1850, so they likely married in the US. 1880 census shows the older children born in Ohio, so he likely settled there first.
1870 Census shows a bunch of children named Carroll with the youngest being 4 and all born in Ohio, and then there's a Margaret, aged 8, born in Missouri, so wonder if she's adopted or something.
Edward's death certificate in 1940 names his mother as Ann Quinn. Both parents born in Maryland (incorrect). Ann's death record in 1912 names her parents as Edward Quinn and Mary Bolland, both of Kings County, Ireland.
I find a marriage of Thomas Carroll and Ann Quinn on 12 November 1853 in Clark, Ohio. Says she lives in this county, and he is more than 21 years of age, and she is more than 18 years of age, and both are unmarried.
Several of the children's baptisms shows up on familysearch.org.Looking at the original records (not just the index entry) might provide more information about the parents; ie, where they were from.