Keyless cars let you do that. Typically, on an automatic, the car decides between starting the engine or just having the radio based on whether you are pressing on the foot brake pedal at the same time or not.
Edit: after reading some Mazda forums, it looks like you can just hold the shift button (if you have an automatic) while on park to make the radio stay on, no need for neutral or other gear. You might want to try that.