You probably aren’t going to be able to control that directly from the permissions, the grant that controls the update is PURCHASE_PART_SUPPLIER_API.MODIFY__ , but that applies to the whole record, not to the Status specifically.

You could turn on history logging for that Status as a way to track who is changing that status.

You could create an event that detects when changes are going to be saved to the record and if the status changes block that save unless it is a validated user for that purpose.  You would need to hold the validated list of users who can change in a Custom Logical Unit for this purpose.  We do this with Document Class selection for example.

Thank you @ShawnBerk 

I am still a novice user but learning  fast.
Could you be so kind as to walk me through your solution. I would highly appreciate it.
And may be some screenshots. Thanks

We added a Custom Logical Unit that holds the relationship for which Logical Unit can have which document Class

The Event Action determines whether there is a match in the Allowed Logical Units

The Event fires when New Objects are created

So you would need the same as a minimum I think, the list of users who are allowed to change the status, the event action to validate against the list, and the event to trigger when an existing SfPP is changed to see if the status has changed.