Any query or form that is derived from your parameter query will ask for those parameters. I'm not sure what you mean by underlying tables being a problem, from what I can tell all is behaving as expected.
To answer your question directly the only way I know to prevent the popup would be to set the parameter's value in vba, but I'm not sure you need to go that route. There's a 100+ ways your user could select a record for editing, the most obvious way would be to create a search form and remove the parameter from the query.
When you open your form you would filter it to a specific record, something like this:
Code:
DoCmd.OpenForm "Copy of qryDocumentVersions", , , "DOCUMENT_NUMBER = '" & some_criteria & "'"