Why doesn't the voltage of the load fluctuate in a Zener Diode circuit?
When using a Zener diode in parallel to regulate voltage, you do not connect the Zener directly to the voltage source. Instead, you connect it with a resistor (or other circuitry) capable of handling the voltage drop and current required. See this example circuit, where \$V_z\$ is the Zener voltage:







simulate this circuit &ndash; Schematic created using CircuitLab

The voltage across \$R_z\$ will be \$(Vcc-V_z)\$, while the voltage across \$R_{load}\$ will be \$V_z\$. The power rating of \$R_z\$ should be at least \$\frac{(Vcc-V_z)^2}{R_z}\$

You are correct in the statement that connecting components in parallel to an (ideal) voltage source results in that same voltage across each parallel branch. But when connecting things in series with a voltage source before parallel branches, then this no longer applies.