How is it possible for an iframe to access its parents DOM?

If the content of the iframe and its parent have the same domain, you can access the parent pages DOM from the iframe by using parent.document.getElement....

However you can't do this cross-domain (not even across different subdomains) as it will result in:

Uncaught DOMException: Blocked a frame with origin "https://example.com" from accessing a cross-origin frame.

Generally, you can't communicate between the two DOMs across domains. However, there is a way to pass messages between the two using the hash portion of the iframe's url. For iframes on the same domain, see oezi's answer.

This might be of some help, and there's plenty of other questions on the topic around here.