Car will start in neutral but not in park. (automatic transmission)

@marty53 the neutral safety switch needs to be replaced. Mazda calls it the park interlock

"Park Switch Not Engaging – How Access, and Repair Park Selector Switch

Important, be sure the Parking brake is full engaged.

Pull off the cup holder assembly, but lifting just under the center storage lid.

Lift off the shift cover.

Exposed will be the shift assembly, the gray cover will need to be removed (I didn’t know how to remove the shift knob, and just worked around it). From the right side, you will need to insert a small screwdriver under the locking clips; I ended up using a flat blade from a detachable screwdriver, as I needed something short, and a small screwdriver. Lift under the front one first, as I found this one to be more difficult to remove, one both sides are lifted; the cover can be removed (might take a bit of effort). Turn the cover sideways if the shift knob was still in place.

Take a look at the front of the shift assembly and a mechanical lever, there should be a small switch, with the key in the Run position, apply the brake, and gently shift the transmission into Neutral, you should be able to see the mechanical lever move away from the switch. Watch the gear selector on the dash to see if the switches are working properly. My problem was the switch wasn’t fully closing when in park. Before making an adjustment to the switch, turn the key to the Off position, you can’t remove the key unless in Park. Using a small needle-nose pliers, gently bend the metal actuator, at the bend, to the right, don’t over bend; you can always try again if you didn’t go far enough. Only bend the round part, as you are looking to increase the span, and bending the straight part may not help." from here