Change the parent of a menu link using the menu module UI.
function moveMenuLink($item, $plid, $menu_name) {
$mlid = $item['mlid'];
$edit = array(
'parent' => $menu_name . ':' . $plid,
);
$this
->drupalPost("admin/structure/menu/item/{$mlid}/edit", $edit, t('Save'));
$this
->assertResponse(200);
}