How to know what are all the possible parameters for a query string for a site?

It is the script/app running on the server that decides what parameters are valid. Unless the app provides such a query mechanism you can't do it. The server has no idea what is valid and what isn't.

Not guaranteed to get you ALL query strings, but it is often helpful to Google "foobar.com/p1& * ".

You will be able to see all the public occurrences of query strings for the foobar.com website.

(As the accepted answer says, there is no general method to access query strings unless the website provides an API.)