The Constitution says a great deal about who should interpret it. Article III, Section 2, Clause 2: "In all Cases affecting Ambassadors, other public Ministers and Consuls, and those in which a State shall be Party, the supreme Court shall have original Jurisdiction. In all the other Cases before mentioned, the supreme Court shall have appellate Jurisdiction, both as to Law and Fact, with such Exceptions, and under such Regulations as the Congress shall make." Jurisdiction is defined as "the power, right, or authority to interpret and apply the law."
The reply above is a non sequitor. It's true that the U.S. Supreme Court has the final authority to interpret and apply the law in cases that come before it -- this is the doctrine of "the law of the case" -- but this is a far cry from claiming that judicial opinions written in such cases apply to every subsequent case in other courts or that the U.S. Supreme Court has exclusive authority to interpret the Constitution.
The notion that judicial opinions are binding in other cases -- what we might ironically call "the law of the opinion" -- has no basis in the text of the Constitution or in its historic context.
"Judicial Supremacy," the notion that the judicial branch (ultimately the U.S. Supreme Court) has the final authority to interpret the Constitution, is likewise without any textual or historic support. It is a doctrine invented by the U.S. Supreme Court in 1958 in a case called Cooper v. Aaron, in which the Court writes the bold-faced lie that it has always held that it was Supreme in the exposition of the Constitution.
That's simply not true; the Emperor has no clothes. Thank God Justice Parker isn't afraid to say so.