What is meant by "3rd party hosting" is the embedding of a picture on Photobucket (or other picture-hosting sites) into another website, such as a blog. I've been doing this sort of thing for a long time, but only now does Photobucket seem to have a problem with it. I suspect that this might be due to a recent change in their terms of service, or perhaps the number of pictures I've uploaded there and embedded here has exceeded an amount that they might normally allow.
It thus looks like I will have to explore some options. I could, as the notice says, "update" my account, which would naturally involve shelling out some bucks. Since I'm not professional either as a blogger or as a photographer, this option does not seem encouraging. I could also delete some embedded pictures and replace them with links to the corresponding Photobucket page. If so, I will start with some of my earlier posts, thus making the changes to posts which are old and thus probably not viewed all that often. I can also upload pictures directly to Blogger, which some of my fellow Blogger users have done.
Whatever I do, it will be in the near future, since I have no plans to stop traveling and taking pictures of the places I visit. One way or another, the readers of Bigfoot's Place will be shown some of the things I see, wherever I go.