Starter will not engage 2000 Buick Century

I bought my 2000 Buick Century with 14K. At 19K, a single incident, the starter would not engage when I turned on the ignition. I tried again and the starter worked fine. At 29K, again a single incident, the same happened again. Now at 68K the same happens approximately once ever 25 to 50 start attempts. New battery so that is not the problem. I have tried serveral things when this happens: turn the ignition key off and on several times, put the gear shift in neutral, put the shift back in park and get out of the car and manually rock it back and forth. I do not know if any of those things are a cause and effect but the car will start. The next time the starter engages fine for another 25-50 times. The condition does not seem related to temperature, whether the engine is hot or cold, etc. I took it to a Buick dealer and of course it started each time of many times tried. The service manager told me they could not diagnose unless it failed to start when at thier shop. This is very frustrating. Does anyone have any experience with this condition. I suspect that it is in the park/neutral switch or the starter but why at 19K, then again at 29K and now much more frequently. Thanks for any suggestions.

Most likely, to me, cause is the neutral safety switch. Next time it happens, leave the car in park but push the lever as far as you can. If that doesn't do it, put it in neutral, turn the key to start and move the gear shift around in neutral.

Thanks for the help! I think that is the case and the next time the starter defaults I will try your suggestions.