It could slip out of park, but most likely would "park lock." This would mean the parking pawl would jam inside other parts in the transmission, making it tough to get your car out of park. Sometimes pushing the car is the only way. Use the parking brake on a hill.
When climbing a steep incline. The automatic transmission will always try to engage a high gear.
When a vehicle does not go into gear when it is shifted from park to drive, it is usually due to transmission issues. If it has a manual transmission the gears may be stripped, if it is automatic then the transmission may have overheated.
Broken "Parking Pawl"?
Yes it is possible, especially if it was not fully in gear and a manual transmission. If it is an automatic, it must be in Park to hold. Always set the parking brake.
The purpose of the automatic transmission is that is shifts for you. You'll want to lock your transmission into your ideal gear for going up and down steep inclines to prevent it from downshifting, but, other than that, you typically let the computer do the shifting for you.
Drive 1 locks the automatic transmission into first gear (preventing an upshift). Very good to use when parked or stopped on a very steep slope and you need to move forward and do not want the transmission to shift. Drive 2 locks the automatic transmission into second gear. This is sometimes referred to as a "snow" drive position, as it causes the engine power to be applied to the wheels much more slowly (via converter slippage within the transmission) minimizing actual wheel-spin when accelerating from a stopped position.
my automatic transmission is stuck in neutral. how do you get it into gear
Reverse
It is called the passing gear.
That would be the second gear of your automatic transmission.
There are several things that can cause your automatic transmission not to go into first gear. The most common cause is low transmission fluid.
If your transmission is manual, you will have to change gear. If it is automatic, you won't.