First, search for your own posts on this website that use photobucket using a similar search to the following pic.
hybiepoo is the photobucket username that should appear in each of your photobucket links.
From the list of returned posts, right click and select open in new tab.
For a single image:
Edit the post and copy the URL from between the IMG tags.
On your postimage page, up the top, select Upload By URL
Paste in the URL and click upload.
Copy the "Hotlink for Forums" link and replace the photobucket IMG/URL tags in your post with the new link. Submit your updated post.
For multiple images it's the same process, except you can paste multiple photobucket links into the upload box one per line.
After the images are uploaded, choose "Hot Link for Forums" on the left hand drop down instead of "Thumbnail for Forums" and you can copy/paste the new links into your post in place of the photobucket ones.