How did you test it? The center lead is the ground for the starter relay. It should only ground in N & P. The two outer leads are the back-up light leads that complete the circuit in R only. To test the neutral switch you need to have your test light attached to a hot lead such as the + on the battery, and touch the center lead in P & N you will get a light. Any other gear it's bad, or no light in P & N it's bad.
So, if the N s/w tests good it's probably your starter relay. That said, you could have a corrosion problem at the firewall plug also, but I would go after the more common issue.