You are all missing the boat here.

This is WHY you can't find the parents -

Facts you should know about the early census records -
All census records [1790 – 1840] prior to the 1850 census ONLY listed the head of household; whether male or female.

NO specific age was stated for any family member
NO place of birth was stated – city, state, or country
NO city, town, or village is stated – only the county; however some census takers listed the township
NO street address was stated
NO marital status was stated – single, married, widowed, or divorced
NO family relationship was stated – brother, sister, cousin, son, daughter, wife, inlaw, etc…
NO occupation was stated
NO parental birthplaces are stated
NO race was stated [but assume “white”]

1850, 1860 & 1870 census records do not show family relationships, marital status or parental birthplaces.

Step children are not enumerated as “step” children
Adopted children are not enumerated as “adopted”
Grand children are not enumerated as “grand children”
Orphaned children were not enumerated as "orphan"

1850 is the 1st census that shows all family members

1880 is the 1st census that shows parental birthplaces +++