This is a pretty common request. I'd love if you could just set another variable/field as the source for range limits out of the box, but until then... 

Here's a script I reuse quite a lot in different variations.

Put the class startDate on your lookup and endDate on the other field and this will dynamically set the min range on the end date field so it will leverage the built-in validation, show errors, block dates on the calendar menu, prevent submission, etc.

  $('.startDate input').on('change', function(){
    // uses start date value or blank if start date is not set
    // adjust the MM/DD/YYYY portion if you're using different formatting in your field
    // the YYYY-MM-DD needs to stay the same because that's how the min/max are set
    var minDate = $(this).val() != '' ? moment($(this).val(),'MM/DD/YYYY').format('YYYY-MM-DD') : '';
    
    // set min date
    var endDate = $('.endDate input');
    endDate.attr('min', minDate);

    // revalidate if field has value (optional)
    if(endDate.val() != '') {
        endDate.parsley().validate();
    }
  });

If you need this to be set on subsequent forms where the lookup isn't running again, you could either manually trigger the change event right after you assign the event handler, or you could split the inner code out to a separate reusable function.

Then, you could also use the "Value not in range" option in the custom error messages tab to set a custom message for your end date field.