trying to understand erratic LEDs behavior
The problem you have is one that is a design problem. You need to realize that the three LEDs in one string may very well have a higher total voltage drop than the three LEDs in the parallel string. The ones with the lower voltage drop may light while the others do not. 

Since you do not have any intentional current limiting scheme to control the current flow through the LEDs in your testing scenario you are depending upon the forward voltage drops across the LEDs to be more than the sourcing power supply. In this scenario the effective internal resistance of the LEDs and the resistance of the wiring harness are being used to limit the current. Operating in this mode the V / I characteristics of the LEDs can shift with time, aging of the LEDs and the current operating temperature of the diode junctions within each LED.

The solution is to provide for a separate driver for each string of three LEDs or to place a resistor in series with each of the three LEDs that are already in series.