OK, so the information has changed!
Now it appears the red 2-pin female connector is actually from the LED light switch, and it provides power to the LEDs. That's why the gender of the connectors seems "backwards" - it is NOT the fan's supply. What DOES supply power to your fan is an adapter (not really a splitter) that connects to a PSU's 4-pin Molex power output and provides two outputs: one is a "replacement" female 4-pin Molex output so you don't "lose" that power source, and the other is a "standard" 3-pin male fan connector that uses only 2 pins and provides power (through a normal-gender connector) to your fan. This is a common adapter that gives the fan a constant 12 VDC supply so the fan always runs at full speed.
Now, with those changes in info, the picture you posted of how you have connected the new PWM fan is wrong! That photo shows the new fan connected to the red female connector that is supposed to be the switched supply for the LEDs! Instead, the new fan's FEMALE connector needs to be plugged into the adapter from the Molex connector. Then its extra male output connector should be taped up so its pins do not contact anything.
Now, as madmatt30 suggests, you could, alternatively, plug your new PWM fan into a mobo fan port and discard the Molex adapter. That makes the mobo's port the fan's supply. BUT your original purpose was to increase cooling of the HDD, meaning that you want the fan to run full speed all the time, anyway. Connecting it to the Molex supply does that. IF instead you connect it to a mobo CHA_FAN port, then you will need to go into BIOS Setup for that port's configuration and change its setting. The default setting is to control the fan's speed according to the temperature of a mobo sensor. You would want to set it, instead, to running full speed all the time. Your choice which way to connect and set.