There is a reason for both sides being connected in parallel and that is that they are all doing the same thing. Putting them in series would create a voltage drop and both LEDs wouldn't be the same brightness.
Wrong!
The outputs of the LM3915 have the current regulated so voltage drops of the LEDs do not change the current.
Since each LED has a different forward voltage then in parallel one will be very bright and might even burn out and the other will be dim or not lighted.
Leds control the brightness with their current, not their voltage so when they are in series then their currents and brightnesses are exactly the same.
My VU meter project is almost the same as yours since one LM3915 drives 20 LEDs in two columns of 10 LEDs. My LEDs are paired in series.
My LEDs are much brighter than yours with a current of 27mA in each LED.
My supply is 7.2V from a rechargeable battery or 9V from the charger so I use a 10 ohm/1W resistor in series with the supply to the LEDs to share the heat.
Since my supply is less than yours then even though my current is higher then my total heat dissipation is almost the same as yours. My LM3915 and my 10 ohm resistor are warm, not extremely hot like yours.
My VU meter does not use a rectifier like yours so it does not have the error of the rectifier at low sound levels. So I made my project extremely sensitive so it can display a pin dropped on the floor in the next room and therefore I added an automatic gain control circuit that adds 20dB to its range and it also displays loud music, TV or my dog barking.
my project has a DOT/BAR mode switch but I like it in the BAR mode.