Purge an url by API Rest

not by REST API , but a quick solution to what you need

<?php
require( './wp-load.php' );

if (isset($_GET['purge_url']) && defined('LSCWP_V')){
  do_action( 'litespeed_purge_url', $_GET['purge_url'] );
  echo "purging " . $_GET['purge_url'];
}

put it at same dir as your wp-config.php , call it a secret name , then access it as https://domain.com/this_file.php?purge_url=/something/something-else/

then it will purge /something/something-else/ on this site

you can additional add some security check like a predefined password , IP lock …etc to this.