Why do parallel resistors have the same voltage across them?

Remember that everywhere in a wire the voltage is the same. On one side, say side $A$ on the left of the parallel configuration, the voltage is the same everywhere, $V_A$. On the other side, say side $B$ on the right, the voltage is the same everywhere, say $V_B$. So the voltage across either resistor is $V_{AB}=V_B-V_A$.