You can check the basics of how to create web services and web applications in this document.
Here, in the web service, a subVI is called as shown below. This subVI performs simple array processing, waits for a specified amount of time, and outputs the start time and end time of the VI execution.
If a POST request (Parallel.vi) containing this subVI is called multiple times in parallel from a client PC, execution will not be processed simultaneously, but will be processed sequentially. If you need to call/execute the contents of the POST request of Prallel.vi at the same time, create multiple POST requests with the same contents. At this time, the same subVI can be used in each POST request.
As shown above, create three POST requests: Parallel.vi, Parallel2.vi, and Parallel3.vi, and execute them simultaneously as shown below to confirm that they operate as expected.

Red, green, and blue correspond to POST requests to Parallel.vi, Parallel2.vi, and Parallel3.vi, respectively, with wait times of 2 seconds, 3 seconds, and 4 seconds specified. The start time of these processes is approximately the same at 21:02:50, and the end time occurs after the specified wait time has elapsed for each.
On the other hand, when calling a subVI multiple times within each POST request or when improving processing speed using a parallel execution mode For Loop, it is necessary to call the subVI within the POST request as shown below, and then perform an asynchronous VI call inside that subVI.
If you need to call VIs asynchronously or execute them in parallel when creating a Web service, you will need to program as above depending on your purpose.