The "beta" photobucket is pretty bad. If you don't have an up-to-date browser with Flash installed, it's probably unusable.
Here's how it works for me (sometimes):
1. In your photobucket library, click on the picture you want to share. That should bring it up in a window by itself.
2. On the right hand side of the page, you'll see a button titled "Image Links". If there's not a menu below it, click to expand.
3. Under "Image Links", you'll see a button marked "HTML code". The button itself will display a bit of the code. Click on that button. It should momentarily say "Copied", then go back to the HTML.
4. If the button said "Copied", all you should need to do is go back to your YT post and paste it in using Ctrl-V (or right-click/Paste, or whatever you use).
5. If the button doesn't say "Copied" when you click it, then you may not have Flash installed. As far as I know, it won't work without Flash. If you right click on the button, you should see something like "About Adobe Flash Player". If you don't, then Flash isn't installed.
6. It didn't seem to work right for me until I collapsed and re-expanded the Image Links menu. You might try that as well.