So your end result is that you want a pair of LEDs to light up at a time? If that's what you are looking for it can probably be done with the existing circuit. It looks like the board has a voltage regulator that feeds supply voltage to the PIC and the LEDs are direclty driven from the PIC's I/O pins through current limiting resistors in the resistor network R1. You probably won't be able to place the two LEDs in each pair in series because the volatge regulator output is like too low. You can place each LED in the pair in parallel with a small maybe 1Ω resistor per LED. It would be nice to know the actual volatge output of the regulator and the value of the resistors in the R1 resistor network.
If you want 16 LEDs to light up individually in sequence, it would be difficult to modify this circuit to do that for you.