Make sure you have parsing links in posts enabled and then use the insert image url. We are like you, "don't allow HTML" but we've had a few youtube videos posted and they work fine.
Well I mean I've seen a forum that you can past just the URL to the Youtube video and it will automatically embed itself without any HTML enabling. You didn't have to insert the video's URL into the Insert Image box.
It might very well be an option that is only available on versions newer than mine.