how to add extra GET parameters to post to url with add_rewrite_rule?

You want to use the add_rewrite_endpoint since what you are concerned with is at the end of the URL.

function wpseo331941_reader_endpoint() {
    add_rewrite_endpoint('readers', EP_PERMALINK | EP_PAGES);
}
add_action('init', 'wpseo331941_reader_endpoint', 10, 0);

//When you want to check if the 'readers' is being used
if ( false !== get_query_var('readers', false) ){
    //....
}

Note that the false !== has to be present, because your query var “readers” will have an empty string as its value.