Consider a simplified case:
simulate this circuit – Schematic created using CircuitLab
Current flows in both R1 and R2. Why does the current not bypass R2 completely?
This is just an illustration to highlight and reinforce what other answers already say. I1 is a current source, so by definition, there must be 10mA of current flowing through it. In order for that current to flow, it must flow through either one or both of the resistors because there is no other path.
By Ohm's law, if there is current flowing in a resistor, then there must be a non-zero voltage drop across it.
But, the resistors are parallel. They must both see the same non-zero voltage.
By Ohm's law again, if there is a non-zero voltage across a resistor, then current must flow in the resistor.
In the example above, I made both resistors the same value to highlight the flaw in the question. But everything I said above would still be true even if R2 was a thousand Ohms, or even a million Ohms.
simulate this circuit
In the second case, almost all but not quite all of the current will flow in R1. The voltage is going to be pretty close to 1V. But if you put 1V across R2, some current must flow (about one microAmpere).