Although you can do it with vanilla javascript, the "best" methods in my opinion are to use either a custom Parsley validator or leverage the min date attribute of the end date field as both mimic the behavior of the built-in error messages, including preventing form submission.

  // Custom validator for date range
  window.Parsley.addValidator('daterangevalidation', {
    validateString: function(value) {
      var allowed = true;

      if($('.startDate input').val() !== '' && $('.endDate input').val() !== ''){
        return Date.parse($('.startDate input').val()) <= Date.parse($('.endDate input').val());
      }

      return allowed;
    },
    messages: {
      en: 'End date must be after start date.',
    }
  });

  // Assign validator to date fields
  $('.startDate input').attr('data-parsley-daterangevalidation','');
  $('.endDate input').attr('data-parsley-daterangevalidation','');

I'd say the better way is to just add a change event to the start date field that modifies the allowable range for the end date.

  $('.startDate input').change(function(){
    var startDate = $('.startDate input').val();
    
    if(startDate != ''){
      var d = new Date(Date.parse(startDate));      
      var dmy = [d.getDate(),d.getMonth() + 1,d.getFullYear()];
      
      // Format date
      for (var n = 0; n < 2; n++){
        if (dmy[n].toString().length < 2){
          dmy[n] = "0" + dmy[n];
        }
      }
      
      $('.endDate input').attr('min',(dmy[2] + "-" + dmy[1] + "-" + dmy[0]));
      
      if($('.dateRange select').val() == 'Between' && $('.endDate input').val() != ''){
        $('.endDate input').parsley().validate();
      }
    }
  });

Which would also give you an error message and prevent submission. Combine that with a custom error message in the Error Messages tab

and you get,