How do I pass an extra parameter to Jquery Autocomplete field?

You need to use a different approach for the source call, like this:

$("#product").autocomplete({
  source: function(request, response) {
    $.getJSON("product_auto_complete.php", { postcode: $('#zipcode').val() }, 
              response);
  },
  minLength: 2,
  select: function(event, ui){
    //action
  }
});

This format lets you pass whatever the value is when it’s run, as opposed to when it’s bound.