I think you can get away with one without that much difference in voltages..
depending on what resistor you are using (in my test i used a 1kohm). I got a reading with the single resistor with a 12v power supply of 1.786v after the R1, if I use two LEDS in parallel I get 1.720v after R1.
NOTE: It is late so I may not be thinking at 100%. I suggest getting a second opinion on this.