Car Shuts Off please help me figure this out

I have a Prius 2013. It's been randomly shutting off every couple of weeks. This is what my car does: I will arrive at the destination, and without any issues lock the car usingk key fob.

1. When it's time to go home, I will use the key fob to open the car. The car responds without issues.
2. Then I attempt to start the car but it is fully dead. It won't start, there are no overhead lights, it's like the car battery is dead. (The battery 12v is new, just changed.)
3. A wait a few minutes, I get out of the car, use the key fob to lock / unlock the car, the car responds and beeps.
4. I start the car but it will only shift to neutral or park, it will not shift to drive or reverse.
5. I will shut off the car and turn it on again and then it starts working like normal.

Every time this happens the car clock will be missing a few minutes (5-15mins) and there are no abnormal lights or codes to let me know that there is an issue.

I can't figure out why this pattern works to get the car working again.

Thank you for your help!!!

 

Thank you for the info. Going to go look at the battery.