There are a number of reasons this could happen. there are neutral safety switches in both the automatic and stick shift versions of most vehicles. There is also a starter relay in the fuse box on some vehicles. check the battery cables to see if they are clean. I have seen people clean the outside of the cables but there needs to be a good connection between the post and cable or side terminal. they need to be removed to be cleaned. You say the starter is good but a bad starter solenoid could cause this problem. If this is the case I recommend replacing the whole starter. Ford cars have a separate solenoid and can be replaced separately.