Benjamin Rush was one of the signers of our Declaration of Independence. One could suppose that men who were so involved in the founding of our nation might have a good idea of what was allowed and not allowed by our form of government.
Today no teacher in a public school would even consider teaching from the Bible, or even allowing daily Bible reading as part of class. Teachers and administrators have stated that such a practice would be against the "separation of church and state" (by which they mean against the Constitution).
Certainly many modern-day parents would not be in favor of such a practice. Even many Christian parents would not want to trust the teaching of the Bible in an environment that has often been hostile to the Bible. But that leaves the question of whether or not it is constitutional to do such a thing.
Consider the opinion of Benjamin Rush. In 1786 he said the following:
I do not mean to exclude books of history, poetry, or even fables from our schools. They may and should be read frequently by our young people, but if the Bible is made to give way to them altogether, I foresee that it will be read in a short time only in churches and in a few years will probably be found only in the offices of magistrates and in courts of justice.
Perhaps Rush might think that, if parents wish the Bible to be excluded from teaching, that the point be made on the basis of logic or desire, but not because doing so would be disallowed by our nation's laws and founding documents.