post_thumbnail_html only for specific thumb size

You’re getting the size as the $size as the 4th argument of your callback function. You just need to use an if statement to check it:

function thumb_bookmark_html( $html, $post_id, $post_image_id, $size ) {
    if ( $size === 'thumb-bookmark' ) {
        $html="<a class="MyClass" href="" . get_permalink( $post_id ) . '" alt="' . esc_attr( get_the_title( $post_id ) ) . '">' . $html . '</a>';
    }

    return $html;   
}
add_filter( 'post_thumbnail_html', 'thumb_bookmark_html', 10, 4 );

Also note that I changed 3 to 4 in the add_filter() function. This ensures that I get all 4 arguments in the callback function.