With a shuttle service:
With a shuttle service, you go straight to the car park and are then transferred to the airport. In this case, keeping your keys is no problem. Only some car parks requested that your keys are kept at the car park, and this is for security purposes only. You car will not be used whilst you are away, save to move it from space to space if absolutely necessary, which is very rare.
You can be sure that the same level of security and attention will be given to your keys as to your car, should you be required to leave them with the car park staff during your stay.
With a meet and greet service:
With a meet and greet service, you are met at the airport and your car is driven back to the car park by a qualified driver. In this situation, it is of course necessary to leave your keys with the car, in the hands of the car park.
If you have requested any additional services, such as a car wash or service, it may necessary for you to leave your keys with the car park for example, to allow the staff to clean the inside of your car).