Your script trigger has the same parameter option as a button so you could explicitly pass the same parameter value. (You could type in the same value as you've typed in for your button.)
I'm guessing that's not what you want here. Don't know of any way to reference the parameter setting for a button from a script.
Yeah, your right; the wish is to have the same script run from multiple buttons using the same script parameter as the button it is run from...