As mentioned in your question, there is support for Flickr in iOS. From UIKit/UIActivity.h:
UIKIT_EXTERN NSString *const UIActivityTypePostToFlickr NS_AVAILABLE_IOS(7_0);
And as the others mentioned, you're free to build your own UIActivity subclass which supports Flickr uploading as well. For that I would suggest the note on subclassing UIActivity in the docs.