Quantum computing essentially relies on the fact that qubits maintain multiple possible states simultaneously. Parallel computing too processes multiple states simultaneously. So what is the difference or how are they different?
For a parallel computer, we need to have one billion different processors. In a quantum computer, a single register can perform a billion computations. This is known as quantum parallelism. Also a quantum computer returns a single randomly-selected correct result, while a parallel computer can directly return all valid results. That's the reason why reading results out of quantum computers generally is slow and getting all results requires running the quantum computation enough times to be relatively certain that the random sampling has seen all possible results.
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See this also: differences-between-quantum-computing-and-parallelism
One big difference is that in parallel computation separate processors need not be executing copies the same instruction at the same time in lockstep; they may be more loosely synchronized or in some cases completely desynchronized. In fact, separate processors need not be executing the same sequence of instructions. This is the distinction between SIMD [Single Instruction, Multiple Data] and MIMD [Multiple Instruction, Multiple Data] parallelism. Quantum computers are by nature SIMD (interpreting quantum superposition as equivalent to processor replication).
Phrased another way: With multi-core you can run all the different processes on a laptop simultaneously; you cannot do that with a quantum processor.