The performance of a parallel distributed detection system is investigated as the number of sensors tends to infinity. It is assumed that the i.i.d. sensor data are quantized locally into m-ary messages and transmitted to the fusion center for Bayesian binary hypothesis testing. Large deviations techniques are employed to show that the equivalence of absolutely optimal and best identical-quantizer systems is not limited to error exponents, but extends to the actual Bayes error probabilities up to a multiplicative constant. This is true as long as the two hypotheses are mutually absolutely continuous; no further assumptions, such as boundedness of second moments of the post-quantization log-likelihood ratio, are needed.