Steve, the anti-parallel diode arrangement is used when the reactance of a capacitor is used to drop the voltage from line voltage to that acceptible for the LED, instead of a resistor. If you didnt have the anti-parallel diode, the cap couldnt discharge, and current would stop flowing, and as you say, a LED could be used instead to use the other half of the waveform. 2 LEDs in anti-parallel and a capacitor in series could be used. If you used a high-efficiency red LED, you could get away with ~2mA instead of 5-10, and a ~40nF cap in series(if I did my math right :p ), though stuff may not fair so well in the case of noise, or if whatever device you have it on gets hooked to a square wave inverter etc...