Prius 2007 switches to neutral or park while in motion

You said you have a couple of codes but only listed one? What's the other code?

As for the photo, that happens when driver's door is open and car is in drive. Not related to your probelm.

Main concept to understand here is car goes into neutral when it has something seriously wrong with it... I suspect both your 12v and your hybrid battery got problems.

 

If the car door is opened when in DRIVE, the CAUTION screen that appears tells you to shift to the P position when parked. Along with a blurb about batteries not charging if the shifter is in Neutral.

 

Since this is often related to a bad 12V the OP should confirm with a voltmeter that the 12V has a decent voltage on it. It is a "new" battery but it might have been on the shelf long enough to fully self discharge, making it a "new" but fully discharged battery, AKA bad. Then verify that both cables are on the posts properly. The clamps should be all the way down, not part way up on the post. Measure the voltage on the posts at the battery, then again at the jump point in the fuse box (with an unpainted bolt on the body for ground). Have one person start the car while another monitors the voltage at the jump post. It should not fall very far. If it drops down below 10.5V (if memory serves) some of the car's electronics won't be happy. If the car starts the voltage will instantly jump up above 13V, possibly as high as 14.3V or so. If the car doesn't start then the battery voltage will slowly return to its original value, but it may take several minutes to do so. That slow recovery is normal, the chemistry in the battery only goes as fast as it goes. The colder it is, the slower that will be.

Another test is to jump start the car with a known good battery. But be exceedingly careful to put the positive lead on the jump post, because doing it backwards will break things. If it starts from a good battery, but not from the one inside, that pretty much means the problem is with the 12V battery or associated wiring.

Finally, the p lock screen can also occur when there is a corrosion problem in the fuse box. The "fix" is to add a wire that gets around the corrosion. Worry about that after definitively showing that the 12V is good all the way to the fuse box.