For all multi-tape Turing machines (i.e. parallel systems) one can create a single-tape Turing machine with a functionally equivalent instruction set architecture. One can apply the Invariance theorem to show that all Turing machines with 1..n tapes are equivalent both functionally and in terms of algorithmic information complexity.
So Southack is essentially correct on this point, and parallelism is not an escape hatch from standard computational systems analysis.
OK, but the point remains that DNA is a tape that is read but not written (here's I'm excluding genetic recombination and similar phenomena, which are not essential processes for all living organisms). As I understand it, the Turing machine doesn't have to write to a tape, but a machine that doesn't write isn't really a computing machine.