<?function is_ajax() {
            return isset($_SERVER['HTTP_X_REQUESTED_WITH']) && strtolower(getenv('HTTP_X_REQUESTED_WITH')) === 'xmlhttprequest';
        }
        // Получение Категорий из БД
        function get_cat_options($parent) {
            $sql_cat = 'SELECT * FROM `cat` WHERE `parent` = '. intval($_REQUEST['cat']) .';';
            $result_select_cat = mysqli_query($on_link, $sql_cat);
            ob_start();
            while ($row = mysqli_fetch_object($result_select_cat)) {?>
                <option value="<?=$row->id?>"><?$row->name?></option>
            <?}
            return ob_get_clean();
        }
        if (!empty($_REQUEST['cat'])) {
            $subcat1 = get_cat_options($_REQUEST['cat']);
            if (is_ajax()) die $subcat1;
        }
        if (!empty($_REQUEST['subcat1'])) {
            $subcat2 = get_cat_options($_REQUEST['subcat1']);
            if (is_ajax()) die $subcat2;
        }?>

        <?// Получение Категорий из БД?>
        <select class="row-item-2" name="cat">
            <option value="0">Категория</option>
            <?=get_cat_options(0)?>
        </select>
        <select class="row-item-2" name="subcat1" style="display:none">
            <option value="0">Подкатегория</option>
            <?!empty($subcat1) ? $subcat1 : ''?>
        </select>
        <select class="row-item-2" name="subcat2" style="display:none">
            <option value="0">Подкатегория</option>
            <?!empty($subcat2) ? $subcat2 : ''?>
        </select>

        <script>
            function get_cat(from, to) {
                $.get(location.href, { cat: $('[name="'+ from +'"]').val() }, res => $('[name="'+ to +'"]').append(res));
            }
            $(document).ready(function() {
                $('[name="cat"]').on('change', () => get_cat('cat', 'subcat1'));
                $('[name="subcat1"]').on('change', () => get_cat('subcat1', 'subcat2'));
            });
        </script>