Does it help to uncheck the "Web part content" checkbox in the right top corner on the design tab of the page you're working on?
You could also go to the "Page Templates" module, select your page template try the design tab over there.
If both didn't work a few options (all in the page template module):
I've seen this error when you've got "Template type" selection on "ASPX + Portal page" on your template. Check the general tab. And check if "Portal page" is selected.
if you have versioning enabled you can try finding a previous version in the Versions tab of the page template.
If this will give you the same error you could try finding the webpart in the "Web parts"-tab in the templates module. This is an XML view of all web parts on that specific template.
BE WARENED, as the message states:
This dialog should only be used for emergency reasons. If you want to modify the web part settings, use the Design mode of the page.
Hope any of these tips help.