Hans Zuercher - Need Help Sorting These Connections

I'm a descendant of the Hans Zuercher who was married to Verena Zwygart and they were the parents of my ancestor Christina (Zuercher) Fridig. The info I've had this far back on their line has been sketchy. I have had Hans' parents as Peter Zürcher 1592- and Anna Schmid 1590- as on this site:

https://gw.geneanet.org/kes?lang=en&p=hans&n=zurcher&oc=9

and as connected here on Geni, but Geni has an extra father connected - Anthoni. Is there documentation for his being Hans' father?

I also see that there are two sisters connected to Hans who are named Anna. They have the same parents - Peter and Ann(a), so those could be merged.

I'm just really needing more info as to which parental connections are correct for Hans. Any additional info / clarification would be appreciated. Thanks in advance for any help.