Here is the best case I can make for Moses Barker being the father of John W Barker. We already know that the only Barker in the 1820 Perry County, IN census was Moses Barker. But let's try to determine where Moses was before and after that. These are the records of the person who was likely the same Moses Barker. Note that the only way our JWB could be his son is if the age of JWB was incorrect when he registered for the Civil War. If he was actually seven years older, the math works. Why would he have lied? Because enlistment cut off at age 45. It is possible he made himself appear younger than he actually was in order to be accepted into service.
Also note that the 1830 census for Moses Barker in Crawford County, IL and the 1840 census for Moses Barker in Knox County, IN seem to line up also. Ages of family members synch up nicely. And...Crawford County and Knox County are adjacent! I think we have a pretty good picture of where Moses was from 1813 on. Now, whether JWB was his son, that I'm not as sure.