Update: here's a simpler method to find and download your Photobucket image.
1) edit the forum post so you can see the IMG information.
2) copy the link inside the IMG tags.
3) open a new browser tab.
4) paste the link in the new browser tab and a ".html" to the link. Hit the ENTER key.
5) that should open up the Photobucket page for that particular image (you may need to log into Photobucket if you haven't already).
6) the image will probably be hidden behind the "please give us money if you want to see your photo" sign, but ignore the warning and just right click on the sign.
7) a menu should open. Choose "Save image as."
8) save the photo to your PC.
9) go to the saved photo, rename it with an intuitive name. Now you can upload it to a hosting site and finish editing your forum post to show the newly hosted photo.
There are probably other ways to do it. For me, the hard part is finding the particular photo I need out of over a thousand photos in my Photobucket account. This method takes me right to the correct photo.