You said parts are gone? If its just a bolt or nut you can just replace the missing parts.
And yes a poor connection can provide enough current to spin the starter, but prevent the correct voltage and current from reaching finicky parts like the ignition and computer systems. We are not talking about a simple ignition system based on points and a coil.
One of the frustrating issues with this board is trying to convince people to actually check their electrical connections before looking at more complicated and costly items.
Use this opportunity to check both ends of all battery cables, big and small, negative and positive. Are the connections tight, clean, free of corrosion, free of rust and protected with dielectric grease? Once you locate these connection points it a simple 2 minute procedure to check them every oil change. Helps to make sure your daughter isn't stranded on a dark and lonely night.