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The answer comes from Energy Conservation Principle which commonly is known as Kirchoff's Voltage Law.

Voltage is basically work done per unit charge and work is form of energy which has to be conserved. Also, electric fields are conservative that means work done depends only on initial and final points.

Combining all that I have said, if you make your starting and end point same, that is run in a loop, effectively work done will be zero (Potential difference is zero). Thus in a parallel circuit if the voltages are not equal, changing the loops will violate this principle.

Let's say you have 2 resistors in parallel attached to a battery. If you assume that the voltages of branches are different, then considering the loop that contains both the resistors and applying KVL violates energy conservation and that's not possible. Thus the only possibility is the voltage has to be same.

Additional Note: The same voltage might not be equal to battery voltage if there is any resistor in path of parallel wires and battery. But the parallel voltages will be equal.

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Related Questions

What happens to a parallel circuit resistance if the voltage applied is doubled?

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