There is no easy answer, but in general, you have it right.
Putting more LEDs in parallel (they are voltage devices, so they all need to see the same voltage) will add to the collective light output - to a point. The point is reached when whatever is supplying voltage to the LEDs can't give any more - that's the larger P you noted. Some remotes can drive more LEDs than others. (Some are brighter than others, too!)
Most IR LEDs are the same, or same enough for this purpose. Instead of hacking apart another remote (which should work OK), go to radio shack, spend the dollar and buy an IR LED brand new. Be sure to get polarity correct!
For teams who want to pump out enough IR light to melt aluminum
you can put a properly designed Op-Amp circuit between the LED output and the bunch-of-LEDs, there are Op-Amps that can drive a dozen or more LEDs easily.
Just be advised that you really will not need a lot of IR light, and excessive light will only interfere with everyone - including reflections that will hurt your own range as well. One or maybe 2 LEDs should be more than enough for the 50 or so feet you'll need it for.
Don