Remote indicator light for alarm panel when armed
You are on the right track! But -- the security system will have a current limiting resistor in series with the LED, and the SSR will have it's own current limiting resistor.  If you try to connect the SSR where the LED connects, those two resistors may add up to too much resistance to operate the SSR. Also, the LED may stop working!
You need to understand how the LED is powered. It will probably have a resistor in series (i.e. connected to one pin of the LED). Call that pin 1. Use a voltmeter to probe from the other LED pin (pin 2) to the far end of the resistor, measuring voltage and polarity with the system armed and disarmed.
Now just get an SSR that activates on that voltage, and preferably one that doesn't take more than a few milliamps. Connect it up. The built-in LED should keep working.