The problem would be fixed when profile pictures would be copied to the Stack Exchange servers and served from there.
Meanwhile, you can take steps to protect yourself from 3rd party tracking cookies by using a browser add-on which blocks them, like Ghostery or Privacy Badger. Also, some web browsers can be configured to only accept cookies from the currently visited website and not from any other websites from which any content is embedded.
For Firefox, there is also the very handy extension Self-Destructing Cookies which allows you to selectively set per website which cookies you accept, reject, or accept at first but delete them when the tab is closed.
However, Facebook will still register the access from your IP address. When you don't want to interact with Facebook at all, you can configure your operating systems
hosts file to route the requests to all Facebook domains to
0.0.0.0, making it practically impossible to access any Facebook services at all from your computer.