I am controlling a stepper motor and an LCD screen from one raspberry pi. Originally, the motor and screen were powered from the 5V output pin on the raspberry pi. I found that this would draw too much power and one or both of the devices wouldn't work.
I have a variable power supply, if I was to set this to 5V can I connect this in either series or parallel to the GPIO pins on the raspberry pi?
Would this cause any damage to the raspberry pi?
(I'm aware that I could just power the Pi from the power supply and that there are other solutions to this get around connecting the PSU to the Pi but that is not what I am asking)
All the advice I have seen says no. Use the Pi's power supply via the microUSB or use your own power supply via the 5V and ground pins. If you use both they will "fight" with each other and cause problems.
Why not power your external kit with the external 5V supply? You can still share the grounds if the external kit has logic which needs controlling from the Pi's GPIO.
DO NOT connect a "variable power supply" to the Pi! Most (laboratory grade supplies excepted) are poorly regulated.
DEFINITELY DO NOT connect any power supply to the 5V rail of a powered Pi. (It is acceptable to power the Pi through the Header pins - provided the power supply meets the Foundation recommendations.) Your confusion between series/parallel is a further concern.
By all means power your stepper from an external supply, but unless it is particularly demanding you should be able to power from the Pi. This indicates your power supply is probably inadequate.