I'm trying to start a python script that will parse a csv file uploaded from the UI by the user.  On the client side, how do I make a call to start the python script (I've read AJAX http requests work)?  And then secondly, how do I take the user input (just a simple user upload with the  HTML tag) which will be read by the python script?
The back end python script works perfectly through the command line, I just need to create a front end for easier use.
Many (if not all) server side technology can solve your problem: CGI, Java Servlet, NodeJS, Python, PHP etc. 
The steps are: 

In browser, upload file via AJAX request.
In server, receive file sent from browser, save it somewhere in server disk.
After file is saved, invoke your python script to handle the file.

As your current script is written by Python, I guess Python is the best choice for server side technology.
Start python script from client side