The cell isn't a Turing machine either, in that there is no 'write' function. If you regard DNA as the tape; it's read, but it's certainly not written. The essence of a Turing machine, as I understand it, is that bits are read and written sequentially. The cell reads many DNA sequences in parallel.
That being said, DNA is my area of expertise, not computer science; I look a grad. course in machines languages and computation 20 years ago, but since then my knowledge of the subject is at a Sci. Am. level.
How can one copy DNA without doing any "writing"?!
Are you saying that cells don't replicate DNA?
Cell Replication
Homologous chromosomes: one from mother, one from father. Humans have 23 of these pairs of chromosomes (46 total chromosomes).
Only certain cells in plants meristems actively dividing
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.