Here's a counterexample. What if the Catholic Church stipulates to its priests they won't be allowed to be recognized as priests if they are gay or pedophiles. A similar argument (false argument) might be held that the leadership of the laity was being held hostage by such a standard.
In this case, the church might need to defend itself from entering into an agreement with a party who in turn, if fails to act within the guidelines understood by the church as minimally acceptable, then the church would expose themselves to liabilities caused by the parent which they never intend to agree to accept.
Church is well within their rights to reject the relationship, IMHO.
I think you are grasping at straws. The two examples you state are mutually exclusive, the church has never railed against someone merely being a homosexual, and the other is clearly a violation of civil law.
What is going here is much like a merchant in a free society stipulating requirements against the law, before transacting business. It is much more like the phone company demanding that you convert to islam, before allowing you to connect to their service. The law should come down hard on this school and demand that they cease and desist, as this parent has committed no crime. If this comes to trial, it is my opinion that the parent will win, and it will be determined that the school never had the right to demand arbitrary behavior beyond what the law currently demands.