This should read "websites that can't parse mobile links for non-mobile users and vice-versa". There's nothing worse* than clicking on a link on your mobile and ending up getting redirected to the homepage of some sites mobile version and not getting the content you wanted.
Basically, yes. For example, I know that a certain large payment company that's owned by a large auction site have a new strategy of "mobile first". All new content/features will be designed for and implemented on mobile platforms before being made available on the desktop site.