That thing between the tranny and carb is your oil pressure switch. I think your linkage is misadjusted and although the trans is in park the neutral start switch in the column thinks it is in gear. You could try disconnecting the trans linkage (make sure it's in park position!!!!) and then try starting the car with the shift lever in park and then neutral while moving the shift lever. My guess is your neutral start switch is shot or the linkage is out of adjustment, the switch is in the column behind the steering wheel. You will have to remove the wheel to access it. It ties into the ignition switch which sends power to the solenoid.
“She may not look like much, but she's got it where it counts, kid. I've made a lot of special modifications myself.”
— Han Solo