Send comment reply emails to the parent comment author automatically in WordPress.

comment_reply_emails.php

<?php
add_action( 'wp_set_comment_status', 'wi_comment_notification', 99, 2 );
function wi_comment_notification( $comment_id, $comment_status ){
	$comment_object = get_comment( $comment_id );
	if( $comment_status == 'approve' && $comment_object->comment_parent > 0 ){
        $comment_parent = get_comment( $comment_object->comment_parent );

        $mailcontent = 'Hi ' . $comment_parent->comment_author . ',<br><br>';
        $mailcontent .= $comment_object->comment_author . ' replied to your comment on <a href="' . get_permalink( $comment_parent->comment_post_ID ) . '">' . get_the_title( $comment_parent->comment_post_ID ).'</a> with the following:';
        $mailcontent .= '<blockquote>' . $comment_object->comment_content . '</blockquote>';
        $mailcontent .= 'See their comment and reply: <a href="' . get_comment_link( $comment_object->comment_ID ) . '">' . get_comment_link( $comment_object->comment_ID ) . '</a><br><br>';

        $headers  = 'MIME-Version: 1.0' . "\r\n";
        $headers .= 'Content-type: text/html; charset=UTF-8' . "\r\n";
        $headers .= 'From: Wired Impact Blog <xxxxx@wiredimpact.com>' . "\r\n";

        wp_mail( $comment_parent->comment_author_email, 'New Reply to Your Blog Comment', $mailcontent, $headers );
    }
}
?>