Proper way to send an Authenticity Token with AJAX to Rails

Actually, you are reading the action attribute of form and sending a post ajax request to it. to send form data you have to submit the form or you can serialize the form data and send it in ajax request like

$(".ajax-referral").click(function(){
  $.ajax({
      type: "POST", 
      url: $(this).parent("form").attr("action") + "?&authenticity_token=" + AUTH_TOKEN, 
      data:$(this).parent("form").serialize(),
      dataType: "script"
      });
  return false;
});

Doing this will serialize your form data and send it along with ajax request and authenticity token is already being sent via query string