The p button in my car is the parking assist button. It is often accompanied by an indicator light that shows the capital letter P and a symbol of a traffic cone to indicate that your vehicle's sensors are actively scanning and looking for objects in front and/or behind your car.
Another p button is a privacy lock. It can be pushed when you want to keep the key fob in your pocket or bag to prevent someone from using the car while you are inside the store, restaurant, or restroom.
The p button can also be used to disable the proximity alarm, if you're in a parking lot and you don't want to hear your car beeping every time a person walks within range of the sensors. This is especially helpful if you're parked and don't want to turn off the ignition, but need to make a phone call, use the restroom, or set navigation points.