it says its allowed on moderated forums... there is a hidden meaning to it.... the admins/mods have to take care that only permissible content is written/shared... if not and if google finds out then you can be banned.
as long as your forum and its posts are not against the TOs of adsense, then placing them there would not be a problem..
This don't depend of the website programming or structure (not at all) depend more of the content of the website. If your forum don't have content against the TOS of Adsense you can place the Ads in any place that you want.
Yes adsense is allowed on forums or any page for that matter as long as it does not violate adsense terms of service. Just make sure you follow the rules laid by Google and you will have no problems at all...