I went looking for Hans Jerg Finter's marriage to Anna Catherine Finter on or about Nov. 2, 1706. I discovered on familysearch.org that there was a German Marriage on the same date between Johann Georg Finter and Anna Catharina Schwartz in Karlsrhue. I think this is his correct name on the basis that "Johann" could easily be nicknamed to "Hans" and "Georg" could easily be pronounced as Jerg. Anna's name only has a change in spelling.
If true, this leads to Hans' father's name as also "Johann Georg Finter" and Anna's father's name as "Johann Jacob Schwartz"