How to parse a shortcode within a shortcode?

The function you are looking for is named do_shortcode (there will also be a function apply_shortcode in WordPress 5.4 that does the same). Codex Page

For your example, this will work:

function content_for_logged_in($atts,$content){
    $logged_in_content = "";
    if(is_user_logged_in()){
        $logged_in_content = do_shortcode($content);
    }
    return $logged_in_content;  
}
add_shortcode('logged-in-content','content_for_logged_in');

Happy Coding!