Character (HD44780 based) LCD backlight is always on
The back light is being powered from the supply to the LCD controller - Pins 1 &amp; 2, this is from the solder blobs marked JI &amp; KI near R9.

To run the back light from pins 15 &amp; 16 you need to unsolder the JI &amp; KI blobs first, then solder the JA or JK links

If you want pin 15 to be +5v use both JA links (pin 16 will be ground)
If you want pin 16 to be +5v use both JK links (pin 15 will be ground)

J stands for jumper - I means internal, K means pin 15 is the Cathode (I know but it is standard to use K as a short form of Cathode), A means pin 15 is the Anode...