Why must the voltage drop over resistors in parallel be the same?
I think you're over-thinking this.
Here's how the voltage across two resistors in parallel is measured:



simulate this circuit &ndash; Schematic created using CircuitLab
Since R1 and R2 are in parallel they are each connected between the same nets, net A and net B. That means both R1 and R2 experience the same voltage, that is the voltage between those nets. The voltmeter also experiences that voltage and displays it as well.
Your misunderstanding could result from misunderstanding the relations between voltage, current and resistance.
I prefer to compare voltage to &quot;pressure&quot;, in the case of parallel resistors, the voltage (pressure) on both is the same.
That voltage (pressure) will make charge / electrons flow through a resistor, compare it to how a liquid or a gas flows through a pipe. In electronics we call that flow of electrons (or charge) current and that current is directly a measure of how many electrons flow through a conductor or a resistor.
If the same voltage is applied to two resistors because they're in parallel (see above) and one resistor has a high value while the other resistor has a low value, then the low value resistor makes it &quot;easy&quot; for the electrons to flow through it so more electrons flow (per second) so the current will be higher (for the low value resistor). For the high value resistor it takes more effort so less current flows.