Hi Troels,
Whch part is running your campaign responses entity ?
This entity is available for both Marketing module in Customer Engagement and Dynamics 365 for Marketing application in its sales module.
However, whatever part is, could you let me know how did you download the import template and how it looks like ?
From my point of view, the key point would be that it seems that "Import Data" button was not available in UCI, and Makerting application has not Web Client interface.
Web Client:

UCI:
From my test, I didn't hint "Export to Excel" button,
with this export, it'll just export a excel with data and column in current View, and I got no success from this exported/downloaded template.
Instead, I downloaded another xlsx template from "Download Template for Import"(in my screenshot 1), in this template, most of fields/columns were contained:

For "Parent Campaign" column, I just populated it with Campaign name in text format,
then I imported the xlsx file in Import Data > Import Data in Web Client,
and in Import from Excel in UCI,
it seems that only UCI would try to let us review field mapping before import.

the template import worked well for both two type of interfaces.


So my suggestion:
1. Download the template with completed columns in Web client(in Customer Engagement).
2. Switch to UCI and review fields mapping in UCI to check whether the Parent Campaign column in excel has been mapped correctly to Parent Campaign/Related Campaign field in CRM.
(its compatible for both CE or Marketing application)
In addition, what your CRM version is ? My working environment is Dynamics 365 Version 1710 (9.1.0.8428) online.
Regards,
Clofly