This problem is starting to annoy me!. (in the sense that I can't help, I mean). I don't use Flickr myself, but you could try posting on this thread and give them some earache, the public visibility of it might help.
Did they give you a direct link to a page to reset the password?. Otherwise I was wondering how you reset it without being logged in.
They sent me an email with a click here to reset password. What the password has to do with it, I dont know! Followed the procedure then went to flickr to try n log in, but useless. Its nowt to do with the password really, just sends me round n round, tried reregistering, change password, rubbish. The email had a respond thing so I had another dig and told em "Goodbye"
Somebody on one site said try imgbb. Went through joining procedure yesterday, we will send you a registering email, live only for 48 hours. They havent sent one...................