How to start an automated jenkins job with variable parameters without using CLI

How to start an automated jenkins job with variable parameters without
using CLI

Scenario:
Jenkins is running on a server to which I do not have access to. I cannot
use jenkins CLI in this case.
I do have access to the jenkins web interface via a url and I can see the
job progress and everything.
There is a build job configured to be run by jenkins. It can be run on an
on demand basis using the web interface.
Requirement:
Currently I can specify the parameters defining this external event in the
jenkins job web interface and start the job. But it is manual ( I have to
do this).
I want to run the job above based on an external event in an automated
fashion, which means:
When external event happens start the build specifying the parameters of
this external event, wait for build to complete, then if another event
happens, repeat.
Question:
Without using jenkins CLI, can I pass the parameters defining the external
event to jenkins url as url POST parameters ?
Conclusion: This way I can monitor the external event and if it happens
send its parameters to jenkins url and start a build that way.