After the Zener breakdown voltage, a small increase in voltage across the diode causes large increase in current through the diode.

So in a way Zener diode in the breakdown region acts like a resistor with very low resistance (zero for the ideal case.)

So to regulate voltage across an LED for example, why don't we just use a small resistance in parallel instead of the Zener diode?

Here small resistance means small compared to the resistor in series like we use in a Zener voltage regulation circuit.