Biocomputing and quantum computing are both relatively novel areas of information processing sciences under the umbrella natural computing established in the late twentieth century. From the practical point of view one can say that in both bio and quantum paradigms, the purpose is to replace the traditional media of computing by an alternative. Biocomputing is based on an appropriate treatment of biomolecules, and quantum computing is based on the physical realization of computation on systems so small that they must be described by using quantum mechanics. The efficiency of the proposed biomolecular computing is based on massive parallelism, which is implementable by already existing technology for small instances. In a sense, also quantum computing involves parallelism. From time to time, there are proposals or attempts to create a uniform approach to both biocomputational and quantum parallelism. The main purpose of this article is the explain why this a very challenging task. For this aim, we present the usual mathematical formalism needed to speak about quantum computing and compare quantum parallelism to its biomolecular counterpart.