Drive Gear

I'm assuming you actually have the gearshift in Park or Neutral right? If so, it's probably just a broken or mis-adjusted "neutral safety switch". This switch tells your engine that your car in part or neutral. Try moving the gearshift around until the indicator shows it's not in a drive gear then try starting it... Of course be extremely careful that it's not actually in a drive gear.

If you're not sure, you will have to remove and check your neutral safety switch or have a trained mechanic do that for you.

The switch is usually connected to the actual gear shift and moves when you move the shifter. That means you'll likely have to access the mechanism underneath the center console.