1995 Ford Thunderbird CAR WON'T START

From time to time, mostly when its cold I will turn the key over and and the engine doesn't turn and the starter is not turning like its not getting the signal. The dashboard lights react normally when I turn the key. I have replaced the Key Lock Cylinder and the Electrical Ignition Switch. I used to have to turn the key over a few times while holding it down for a few seconds and it will start. Now its nothing. I have had a problem recently with high reving in park so I would turn the car off in Gear. Is it possible it could cause damage to the Park/Neutral switch to fail? I can still start the car if I put my key at run position and touch the remote starter wire to the 12V wire on the starter. Could the factory alarm also cause the car not to start because I seen a loose wire going to the ignition switch? The alarm had no fuse in it and it was disconnected or bypassed before I had this problem.