SSSasha

Вот решение только вместо марки и модели автомобиля - страна и город в ней, обьединены data-id

$('.header-select[name="country"]').on('change', function () {
      let countryOptionSelected = $(
        '.header-select[name="country"] > option:selected'
      );
      let cityOption = $('.header-select[name="city"] > option');

      cityOption.each(function () {
        if ($(this).data('id') == countryOptionSelected.data('id')) {
          $(this).addClass('show');
        } else {
          $(this).removeClass('show');
        }
      });

      $('.header-select[name="city"] option').prop('selected', false);
      $('.header-select[name="city"] .show:first').prop('selected', true);

      if (countryOptionSelected.hasClass('default')) {
        cityOption.each(function () {
          if ($(this).hasClass('show') == false) {
            $(this).addClass('show');
          }
        });
      }
    });

<select class="header-select" name="country">
       <option class="default" value="">Country</option>
       <option data-id="16" value="austria">Austria</option>
       <option data-id="8" value="germany">Germany</option>
        <option data-id="23" value="switzerland">Switzerland</option>
</select>
<select class="header-select" name="city">
        <option class="show" value="">All cities</option>
         <option class="show" data-id="16" value="wien">Wien</option>
          <option class="show" data-id="8" value="berlin">Berlin</option>
          <option class="show" data-id="8" value="dresden">Dresden</option>
           <option class="show" data-id="8" value="hamburg">Hamburg</option>
           <option class="show" data-id="23" value="bremgarten">Bremgarten</option>
</select>

<style>
select[name='city'] {
    option {
      display: none;

      &.show {
        display: block;
      }
    }
  }
</style>