  We consider tensor-network stabilizer codes and show that their
tensor-network decoder has the property that independent logical qubits can be
decoded in parallel. As long as the error rate is below threshold, we show that
this parallel decoder is essentially optimal. As an application, we verify this
for the max-rate holographic Steane (heptagon) code. For holographic codes this
tensor-network decoder was shown to be efficient with complexity polynomial in
n, the number of physical qubits. Here we show that, by using the parallel
decoding scheme, the complexity is also linear in k, the number of logical
qubits. Because the tensor-network contraction is computationally efficient,
this allows us to exactly contract tensor networks corresponding to codes with
up to half a million qubits. Finally, we calculate the bulk threshold (the
threshold for logical qubits a fixed distance from the code centre) under
depolarizing noise for the max-rate holographic Steane code to be 9.4%.
