Judging from the photo you did NOT buy a starter solenoid, but rather a starter RELAY. The solenoid is integral with the starter. HERE is how that works
Notice there are two "push on" flag terminals, the two smaller wires at one end. One of them gets 12V from the key in "start" and the other terminal must be grounded to fire the relay. THAT terminal is grounded by the neutral safety switch on the automatic, the center terminal, only when the trans is in park or neutral. If you have a stick, that terminal is grounded by pushing in the clutch and activating a switch on the pedal linkage
First thing to do is hold the key to start while working the shifter from park through neutral and back and see if you get a crank or at least a click. If not..........
Dig up a test lamp or multimeter. Identify the wires (which change color over the years) see if one is yellowish and the other dark. The dark one is likely the neutral grounding switch wire. Pull that one off and hook your test lamp to the relay terminal you uncoverered. Twist the key and see if you get a light. if not change the test lamp to the other wire still connected to the relay. If it does not light in "start" for "to be sure" move your light to the remaining wire---you pulled loose. If no light there either, that is on either wire, then the circuit from there, through the bulkhead connector, to the ignition switch or the switch itself is faulty
If you DO get a light on either wire in the above test, hook that wire back up to the relay. "Rig" a jumper clip wire to the opposite flag terminal and ground it. Be careful here that the car is out of gear, or in park. See if it starts. If so, you troubles in the neutral safetly switch or wiring. This is just one wire that goes to the CENTER of a 3 wire connector. Make sure the connector is tight. Do the wiggle test on the shifter while twisting the key to be sure. If no go, check that wire, and if has continuity, replace the siwtch
