The root cause of this issue is Field-Level Security on the Parent Case field.
To ensure visibility is set for the Parent Case field:
1. Navigate to Setup > Customize > Cases > Fields.
2. Click on the field label for Parent Case.
3. Click the Set Field-Level Security button.
4. Ensure the Visible checkbox is set for the affected profiles and click Save.
5. Reload the Find Duplicates page on any case.
Note: You will have the option to select the desired Parent Case after selecting 2 or more cases and clicking the Merge Selected button.