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Moore's law describes the trend that performance of chips effectively doubles due to the addition of more transistors to a circuit board.
Since devices are not increasing in size (if anything the reverse is true) then clearly the space for these additional transistors only becomes available due to chip technology becoming smaller and manufacturing becoming ever better.
At somepoint however you reach the stage where transistors cannot be minimized any further. It also becomes impossible to increase the size of chips beyond a certain point due to the amount of heat generated and the manufacturing costs involved.
These limits necessitate a means of increasing performance beyond simply producing more complex chips.
One such method is to employ cheaper and less complex chips in parallel architectures, another is to move away from the traditional integrated chip to something like quantum computing - which by it's very definition is parallel processing.
It's worth noting that the title of this question relates more to the observed results of the law (performance increase) rather than the actual law itself which was largely an observation about transistor count.
Moore's law necessitates parallel computing because Moore's law is on the verge of/is dead. So taking that into consideration, if it is becoming harder and harder to cram transistors onto an IC (due to some of the reasons noted elsewhere) then the remaining options are to add more processors ala Parallel processing or go Quantum.
I honestly don't really know, but my guess would be that transistors at some point could get no smaller requiring processing power to be spread out in parallel.
It's because we're all addicted to increasing speed in our processors. Years of conditioning have led us to expect more processing power, year after year. But the physical constraints caused by densely packed transistors have finally put a limit on clock speeds, so increases have to come from a different perspective.
It doesn't have to be this way. The success of the Intel Atom processor shows that processors could just get smaller and cheaper instead. The processor companies will try to keep us on the "bigger, faster" treadmill though, to keep their profits up. And we'll be willing participants, because we'll always find a way to use more power.