Create a separate PHP script that does nothing but call these other PHP scripts into action. Then, in your "requested" script, to start the other PHP scripts, you would do something like this:
exec('/path/to/php -f make_remote_requests.php &');
Basically, what you'd be doing is executing a system command to open PHP in a new thread and execute a PHP script that calls these remote scripts into actions. The '&' on the end of the command forks this new process into the background, so that the current script isn't waiting for the separate process to finish before it continues on.