One method of research to get through a brick wall is to see what the typical naming conventions of the area and time were.  In the 1800’s the naming patterns were:

• First son: Named for the father’s father
• Second son: Named for the mother’s father.
• Third Son: Named for the father or the father’s paternal grandfather.
• Fourth Son: Named after the father’s oldest brother or mother’s paternal grandfather.
• Fifth son: Named after mother’s eldest brother or father’s maternal grandfather.
• Sixth son: Named after father’s second oldest brother or for mother’s maternal grandfather.
• Joseph’s boys were: Jackson, Joseph, William, John, Jesse.

Therefore:

• Jackson- Joseph’s father
• Joseph- Anna’s dad (This is correct)
• William– Father’s name (Not correct) or Joseph’s grandfather (we won’t know for a while)
  
• John– Anna’s oldest brother (Freeman) or Joseph’s maternal grandfather (we won’t know for a while)
  
• Jesse– Joseph’s second oldest brother (Not corect) or Anna’s maternal grandfather

Based on this approach, Joseph’s father is named Jackson.  This isn’t conclusive, and there does not appear to be a Jackson Phillips living in the area Joseph is living.  It’s something to consider if it turns out the rest of the naming patterns are correct.  After a cursory search, it appears that Jackson Phillips isn’t located in the 1820 census.