Is Parent Company a custom field on your Phone Call form or maybe you might be referring to the Regarding or Recipient fields?
If it is a custom field it could only be populated with a single value for each Phone Call record that created - which wouldn't work if all the customers are different.
If you mean the Regarding field, when the Phone Call is created via a Quick Campaign the only behavior is to populate that with the Quick Campaign record so it can be tracked back to where it originated from.
If the Recipient field, this field won't show up in the view as this is a special field that has the potential to reference multiple records. Out of the box you would need to drill into each record to see who the Recipient is. One alternative would be to create a custom text field and create a workflow to copy the value from the Recipient field into it. Once added to the My Activities view, it would show a semi-colon delimited list of recipients.