You could have a fault with the car’s wiring. The starter-motor takes its power from the fat, red and black leads that attach to the battery, but it also needs a low-voltage signal from the ignition key that tells it to do its thing. If there’s any sort of loose or missing wire, you might not be making the necessary electrical connections. You could also find that there’s an on-board security system that is blocking the starter-motor from turning. You could even find that the ignition key has a flat battery and is not unlocking the car’s on-board computer.

You say that the battery has voltage, but I wonder how you checked this. You can often put a multimeter across the battery terminals and see 12 volts, but this can be what’s called a surface charge that’s strong enough to light up the dashboard, but then disappears when you put the battery under load (by trying to start the car). Attach the multimeter and check the voltage while actually trying to start the engine. You may find you have a dead battery after all.

When you look at how the Park setting works in an automatic transmission, you can see how this could happen. Basically, Park is engaged by a small pin that (typically) slides into the transmission’s output shaft and locks it, preventing the shaft from turning and effectively locking the drive wheels.

The problem is that the Park pin (also called a pawl) is almost comically small for the job it has to do. The point being that Park pawls have been known to fail, at which point, the car will be free to move if the park brake isn’t applied.

Best practice when parking an automatic car is to stop where you want to park, but leave the transmission in Drive and apply the foot brake. Then apply the park-brake, take your foot off the brake and let the park brake take the weight of the car. Then select Park. You’ll still have the security of being in Park, but you won’t have the car’s weight resting on that little park pawl.

You might be lucky and simply have a low transmission fluid level thanks to a leak, or perhaps even a selector that is not properly connected to the transmission.

But the fact that the transmission works in Reverse but not Drive suggests something internally wrong. You could be looking at a broken torque converter, failed clutch or band, a faulty pump, blocked valve body or even a broken gear set. A transmission specialist is going to be your best bet here.