add the following code at line 1 of your page, this will save the parent and child query string variables to a session and reset the querystrings to use the session value if the querystring is missing:
<?php
if(!session_id()) session_start();
if(isset($_GET['parent'])) $_SESSION['parent'] = $_GET['parent'];
if(isset($_GET['child'])) $_SESSION['child'] = $_GET['child'];
if(!isset($_GET['parent']) && isset($_SESSION['parent'])) $_GET['parent'] = $_SESSION['parent'];
if(!isset($_GET['child'])&& isset($_SESSION['child'])) $_GET['child'] = $_SESSION['child'];
?>