Final answer:
The Lemon v. Kurtzman case established the Lemon Test, a three-part criterion to evaluate the constitutionality of state aid to parochial schools, focusing on secular purpose, primary effect, and entanglement with religion.
Step-by-step explanation:
The Supreme Court case that established three rules for determining whether state aid to parochial schools is constitutional is Lemon v. Kurtzman. This landmark decision introduced a three-part test known as the Lemon Test, which is used to determine if a law violates the Establishment Clause of the First Amendment. The three criteria are: the law must have a secular purpose; it must not advance or inhibit religion as its primary effect; and it must not result in an excessive entanglement of government with religion. The Supreme Court found the Pennsylvania and Rhode Island laws at issue did not adhere to these standards due to the excessive 'entanglement' caused by the state's involvement in religious schools.
Later, the Supreme Court ruling in Agostini v. Felton modified the application of the Lemon Test, allowing for federal aid to pay for public school teachers to teach secular subjects in private religious schools. It is to be noted that the Lemon Test has undergone various interpretations and applications over the years, reflecting the Court's evolving views on the balance between government funding for private religious schools and adherence to the Establishment Clause.