I look at is as a difference in relationship.
You 'part with' something that you possess: "I could be persuaded to part with this pocket watch for $30".
I could be persuaded to part with this pocket watch for $30".
You 'part from' something or a group: "I must part from you for now but I count the hours until I return".
I must part from you for now but I count the hours until I return".