The plugin “BBCode Deluxe” was the culprit for the stripping the paragraph breaks, even when a user did a double return. Deactivating returned the paragraph breaks.
The plugin was the attempt to add more formatting options and BBCode to the forums (See this thread).
Forum posts will now appear as they look when creating a post. If you single-return, then there won’t be a space between paragraphs. If you double-return, then there will be a line break between paragraphs.