Yeah, lets start with what wired in Parallel and Series really means with LEDs first.
Parallel - Both LED Anodes (the +) together and both Cathodes (the -) together is Parallel wiring. That means each one gets the full 5v, but they draw twice as much current (need more power) than just a single LED would. So you're giving each LED 5v, not 2.5v, and each one needs a certain amount of current to work and that adds up with the more LEDs you add in Parallel.
Example: You have that 5v source there from the controller. Those 2 LEDs in Parallel is pretty much the same for Voltage as a single LED is, each will get 5v from the controller, but the Current draw adds up. If you use a single Resistor on each LED (the correct way of doing it) then the value is based on the 5v and the current draw of the single LED. If you would want to use 1 Resistor for both LEDs in that Parallel setup you'd have to calculate it based on the combined current draw of both LEDs. That's not recommended as one LED can fry or open up and throw the whole thing off.
Series - If you wanted to give each LED half the voltage from a higher source like that, then you'd have to wire them in series to make the LEDs voltage requirement higher. One LED Anode to the other ones Cathode, + to -, like batteries in a flashlight, so you have a single + and - connection for the 2 LEDs to go to the power/ground connections. They also draw the same amount of current as just 1 LED in this setup, but that method isn't recommended as each LED isn't exactly the same, so one is going to get slightly more current than the other one and it can fail sooner, but it can be done that way. A Resistor still applies here also (it always should with LEDs) but it's based on the added voltage of the LEDs instead of the single value. How you have it wired right now is Parallel, not Series.
Example: The same 5v source, the same 2 LEDs, but since they are in series the voltage needed for them adds up. (say 2v per LED for this example) so you effectively have a single 4v LED there. The current stays the same in this series setup, so that 4v and whatever the single LED current value are what are used to calculate the Resistor value. This should only be done with LEDs of the same ratings and from the same batch. So if you get 6 LEDs rated at 2v all at once you could series them up to run off a 12v power source, since 2v x 6 = 12, but wiring them in Parallel with each one having it's own Resistor is still the best method for that kind of thing.
The formula for figuring out the Resistor value is R = (VS - VL) / I
R - Resistor (The value of the Resistor in Ohms)
VS - Voltage Souce (The power supply for your LEDs, 5v in the case of that wired controller)
VL - LED Forward Voltage (Depends on the color and how the LED was made)
I - Current the LED needs (Which is given in mA (milliAmps) for LEDs, so divide by 1000 to get the right number to use here, Example: For a 20mA LED you'd use 0.02)
So for that controller's 5v and a single Resistor that's, lets say 2v, and rated at 20mA, using that formula you get this..
Resistor value = (5vs - 2.1vl) / 0.02mA
5v - 2.1v = 2.9v
2.9v / 0.02mA = 145
So 145 for your Resistor value in Ohms, which is closest to the standard 150ohm value. Always go up to the closest value sized Resistor, going down will give it more current than it can handle and shorten it's lifespan.
To figure out what Watt rating the Resistor should be, you use the formula W = A x V
W - Watts (Power rating)
A - Current (The mA rating from the LED)
V - Voltage (The voltage drop across the Resistor, or what's left over after subtracting the LED voltage from the source voltage)
So fr0m that above formula you then get..
Watts = 0.02mA x 2.9v
0.02 x 2.9 = 0.058Watts or 58milliWatts
A 1W Resistor is 1000mW (1000 milliWatts), so it's more than big enough, as are a 1/4W (0.25 or 250mW) or even a 1/8W (0.125 or 125mW) since you only need something that can handle 0.058W in this example. Here also, the next highest rating should be used as a lower one will burn up sooner.
Have a read here for some more info on some of it, and if Google can't help find what you're looking for, then you're probably not looking for help on Google. http://www.kpsec.freeuk.com/components/led.htm
Don't let all that math discourage you either, there are loads of online calculators that will do all of that for you if you want. This is just one of the many, many examples out there. http://led.linear1.org/1led.wiz