Yes, normally from the time you start a script until the time it ends you have at least one process running. But the script can spawn more processes or threads.
However, though I didn't look into this, I would say that if you start a script from a webserver because of a user request the script could, for example, be spawned as a thread from the webserver's process, or run as part of the webserver's process. So to give precise information one would need to know what exactly you are running.
Update: From your comment it seems it may be helpful to look into a solution where you have one process running, which then spawns threads for each request. This can improve speed a lot, but it comes at a cost (for example threads only run as long as the parent runs).
Also you might want to look into queuing the connections. In simple terms, your script can handle multiple connections and goes through each handler checking if the remote server already responded, if not it goes to the next etc. This is quite typical in handling tcp connections and you only require one process and don't need threads.
Anyways, discussion of actual programmatic solutions falls outside the scope of server administration and enters the realm of programming.