Yes, this vehicle shifts into neutral automatically when it comes to a stop in park. I believe it's a fuel saving feature.
If I were you, I would just do a fluid exchange. It's not that hard to do. The factory fluid in these is as black as used oil after even 50,000 miles. I've done 2 fluid exchange, once at 75kish and another at 150k ish and mine still shifts just fine, so it can be reliable.
Please do not use the vent port to add fluid. That port gets full of sludge and you would just be washing it back into the transmission. These do not have a serviceable filter so you do not want to do any type of flush at all, just exchange the fluid.
The procedure really is just to first remove the fill port which is located on the top of the transmission, just look under the intake/throttle area you'll see a small torx or hex next to a badge. Once that's open, you know you are able to fill fluid back in (always do this before draining fluid). Then drop the bottom tray from the car by pulling out the push pins around the edge (some guys cut a hole in it, I dropped mine because I like my car to stay in good condition). You'll see the drain plug on the bottom which is a larger hex bolt with a smaller torx inside it. It's basically a drain tube which sets the level, you'll see when you remove it. Remove the larger one to drain the fluid. Replace it and fill it back up with 3-4 quarts of fluid. Run through the gears, then drain it again, then replace 3-4 quarts, run through the gears, then drain it again and fill it with 3-4 quarts. At this time, get the car up to operating temperature by driving it. Go back in the shop and this time open the small inner screw and with the car running, wait for the fluid to stop dripping. Once it stops dripping put the screw back in, add 1/2 quart, and put it all back together.