The arguments about religion stem from the Constitution which prohibits any attempt to either "establish" a religion, or hamper the "free exercise" thereof.
Most of the arguments like the parent you quoted are garbage (edited) and have nothing to do with the constitution, and are therefore of little effect.
However, is a private school requires students to say a prayer, that is as if the school is saying that everyone must believe in this way. As a result, it is unconstitutional.
The pledge of allegiance is non-sectarian though it does refer to one nation "under God." However, that has been construed to mean whatever deity you believe in, and, children are not forced to say the words, only to be respectful. Of course, if a kid does not want to say the pledge, he will probably take a lot of **** anyway.
The really biggest issue is over school funding. Giving funds to a parochial school smacks of supporting or establishing a religion, and has been the major battle ground.
Then of course there is the interplay between free speech and freedom of religious expression.
People who advocate to put religion in the school are doing so on religious grounds, in an attempt to foster their religious view. That is exactly what the constitution forbids, yet those are the people most apt to claim that the constitution allows it.