1996 Ford Taurus Ignition problem.

1996 Ford Taurus GL, 72,000 miles, Auto. Vulcan V6.

I have this really troublesome problem. Basically my car refuses to start once in a while. This problem happens at random. Doesn't matter if its hot or cold, or how long the car has been sitting. It simply refuses to start once in a while. When I turn the key on, the engine doesn't even crank, or cranks very slowly. Sometimes I requires that I crank it a few times before it eventually turns over. Once in a while, when dealing with this problem, I've noticed that the clock resets, and the lights fail to come on until the car starts. I can always get it running if I get someone to jump the car though.

I have replaced the battery, and it is currently holding a charge of 12.8 volts. The battery is being charged at 30 to 34 amps from the alternator. There is a aftermarket security system with remote start installed on the car. I bought it used and I don't know who installed it, or what brand it is. I suspect that maybe there is a kill switch interfering, or maybe the neutral switch is bad. No amount driving changes this problems occurence. The battery never seems to hold a charge bigger than 12 to 13 volts. Is there anything specific I should be looking at?