Automatic Transmission Floor Shift problems of the 2008 Mazda MAZDA3

I parked my car on numerous occasions in the last few months where the transmission failed to engage in park. It required multiple tries to get the park indicator light on the dash to reflect that the car was safely engaged in park. Before I realized what was happening with the transmission. On at least 3 occasions I was unable to start my car at all when I returned from a brief stop. The problem with failing to start happened when the car was on a slight incline and I had engaged the parking brake. After jiggling the automatic transmission lever repeatedly, moving the steering wheel and applying the brake vigorously, I was eventually able to get the car to start. I'm not sure whether it was rocking of the vehicle from frustration or jiggling the gear selector that finally worked. The car has a new battery and has had all scheduled maintenance performed by my local Mazda dealer. When this happens, other electrical equipment functions but there is absolutely no response from the ignition system.