Flickr uses Yahoo! Updates tool that allows you to connect your Facebook account to your Flickr account. Connecting your Facebook and Flickr account lets you post images to both Flickr as well as Facebook account at the same time. Here's a step by step process to connect your Facebook account with your Flickr account:
Login to your Flickr account and click on 'You' menu.
Now select 'Your account' and click 'Sharing & Extending' tab.
Now click 'Connect' next to Facebook icon.
Click 'Allow' on the pop-up window and clear the box next to any sharing option you don't want to enable. For example: posting Facebook activity on Yahoo! and Flickr, or using Facebook profile picture on your Yahoo! feed.
Click 'Continue' to finish linking your Facebook account to Flickr.
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