it's 30mA when LED's are connected in series and 60mA in parallel
Yes for white and blue LED color.
A1 + A2 + resistor to 5V and C1 + C2 + Ground
Yes, this is called: Connected in parallel
5V is connected to A1, then C1 with A2 and finally C2 with the ground
This is "connected in series"
As a designer you can choose between series and parallel connection, however there are some side conditions:
1) The forward voltage must be lower than the power supply.
This is a must. Otherwise, the LED will not reliabled turned on.
For the white display, connected in series, the forward voltage is 6.4 (see page 2 of the dogm132 data sheet). It can not be connected to 5V or 3.3V of the Arduino Board. Both power supplies provide too less voltage.
2) The forward voltage should be as close to the power supply as possible
This will reduce the power consumption to a minimum. The power consumption for the LED is fixed. It is forward voltage * current (P=U*I). The power consumption by the resistor is P = R*I*I, with I the current for the display. So a low resitor will minimize the power consumption. The resistor value gets low, when the forward voltage is close to power supply voltage.
However, for the design of the above mentioned shield, there was another side condition, which was more important than 2):
3) Is it possible to choose a connection so that the current limitation can be done with a resistor for ALL backlight options?
This will simplify the design of the shield and still gives the user the option to choose between all available colors.
The answer is: Use 5V power supply with parallel connection. In this case, you will find a fitting resistor value for any of the backlight options.
However, now power consumption might be bad in some cases. For example with the amber backlight, the dogm132 shield requires a 32 Ohm resistor and will consume 80 mA. The series connection would reduce current to 40 mA, which means that only half of the power consumption would be required.