You could add a hidden field to form for the page value and add an ID to paginator. When you click a pagination link you update the "page" field in form and submit form.
$('#paginator a').click(function(){
var $form=$('#third2')
$form.find('[name="page"]').val( $(this).text());
$form.submit();
return false;
});
You would also need to add event handlers to all the form fields to change the page back to 1 if anything is changed in form
$('form#third2 :input').change(){
$(this).closest('form').find('[name="page"]').val(1);
});