13/27 doesn't scale. In the set of all parents with two children, where at least one of which is a boy, it is assured that a boy is born on a particular day of the week.

If those parents each came up to you individually and provided you the statement, "I have two children, one of whom is a boy born on [day of week]," then every single parent would be assigned a probability of 13/27 for the other child to be a boy. (Assume that with a large enough sample size, the cluster of parents with two boys born on separate days of the week would evenly split their decision of which day of the week to inform you of.)

The analysis given argues that each case would have a 13/27ths probability of the child you were not informed of being a boy. Yet the group as a whole is determined to only consist of 33.33% parents with two boys.

On an individual basis, if the group was a perfect sample containing 189 cases, split evenly among the 7 days of the week, then 91 cases would be classified as having 2 boys using a 13/27 probability, when in actuality only 63 cases would have two boys.

Since the individual probability that has been assigned doesn't scale to the known probability of the entire group, then it must be incorrect. The probability the other child is a boy is 1/3, the day of the week is meaningless information.