Not quite...
There is no direct correlation between the settings of the buttons and what you are describing as the "driving modes" although the buttons do influence the way in which the control systems choose between those "modes".
EV mode is when all the motive power is coming from the battery and the petrol engine is not running. In normal operation at temperatures above about 10 degrees, speeds below 60mph (more or less) and with a reasonably charged battery, the car's control systems will always run in EV for preference if you don't touch the "Charge" and "Save" buttons.
The car's natural behaviour is to run the battery down to its low threshold running in EV mode, then switch to either Parallel or Series Mode depending on the speed you are doing. Parallel mode is when the car is being driven exclusively by its electric motors with the petrol engine running to drive the generator and provide power to the motors. Serial Mode is when the petrol engine is running and coupled mechanically to the front wheels - most of the drive to the car is direct mechanical drive, but the electric motors can still contribute.
In both Parallel and Serial Mode, there may be sufficient spare power coming out of the petrol engine to both move the car and to top up the battery - if you don't touch either the Charge or Save buttons, you will be running in a hysteresis cycle where the car runs on petrol for a few miles, then switches to EV for a mile or two to use up the charge that has just been built up in the battery.
The Charge and Save buttons really just modify that default behaviour by moving the start and end points of the hysteresis cycle. When you press the Save button, you tell the car to move the low point of the hysteresis cycle from its default value of about 10 to 15 % charge up to whatever level of charge you have when you press the button - you are asking the car to maintain the current level of battery charge. It will still run in the cycle of petrol for a few miles and EV for slightly fewer miles, but will not wait until the battery has gone flat. Pressing the Charge button moves the low point of the hysteresis cycle right up to somewhere close to fully charged so that the car will run the petrol engine pretty much continuously until the battery is recharged.
In all cases, the choice between Parallel and Serial Mode is determined primarily by the road speed, not the buttons - the PHEV does not have a multi-ratio gearbox, so it can only run with the engine mechanically coupled to the wheels at speeds equivalent to the top gear on a conventional car. The settings of the buttons only influence the ratio of EV to non-EV operation, they have limited effect on the ratio of Parallel to Serial operation.
