This seems to be something that crops up from time to time and relates to the AJAX Error issue encountered by @sico7 and @MrJKConnor on this support question. I should have updated the thread with the results of what I discovered on the system that @sico7 is running under, but just as I was about to hit reply, the topic was closed by the remarkably vigilant moderator!
Anyway, it appears that on some hosts (academic ones, for example) where WordPress is not given sufficient privileges to modify the database structure, CommentPress is not able to add a 'comment_signature' column to the comments table of the site it is being activated on. Therefore comments left on paragraphs are never associated with them. One of the symptoms of this is that the AJAX error they report is triggered.
If this describe the symptoms you're seeing and you're using CommentPress on an academic system, it's likely you will need to talk to your server admin(s) to see what can be done. If there is indeed no 'comment_signature' column, they will have to add one manually. The following should be enough:
ALTER TABLE '$wpdb->comments' ADD 'comment_signature' VARCHAR(255) NULL;
$wpdb->comments is the full name of the comments table on your site.