It’s confusing because it’s incorrect. Quantum computers do not perform parallel computation. Performing that many computations, according the laws of physics as we know it (see Landauer’s Principle), would consume more energy than the mass of the sun.
I’m not well-versed in quantum computing, but my understanding is that you set up a quantum system so that all non-solutions to the problem negatively interfere with each other. Therefore the only thing that’s left is the solution, at least with high probability. In particular, a quantum computer can compute Fourier Transforms in polynomial time. This is why they’re a threat to encryption algorithms like RSA and El Gamal, for which there exist polynomial-time reductions to the problem of computing a Fourier transform.