illinois wasn't recording births that early in time.
I'd look for his marriage license. Both sets of parents might be on it. However Christian County was created Feb 15, 1839.
If the parents names aren't on the marriage license, I'd look for the newspaper announcement of their marriage.
Illinois wasn't recording deaths when William died. However I'd contact the Christian County Genealogical Society and ask if they have his newspaper death notice. His parents names might be in it.
If you know when Abram/Abraham died in 1865, you can order a copy of his probate records or look for those in FamilySearch, Catalog. Successful SEarching!