The issue I think is that the Cloud Code calling convention requires you use special Parse headers, not just keys: it may be different if its being called from a browser with sets the referer headers. I'm not sure you'll be able to call it this way directly from Mailgun: you may need a proxy of some sort.
EDIT: I think you'll need to use the Express Webhook implementation instead, and then you can use standard basic authentication. Cloud Code is really for cases where you have control over the HTTP client you're using.