Actually, the "Old Regime in Iran" would be the Shah - the Mullahs are nothing but wet-behind-the-ears, rabble-rousing terrorists. And even a government led by a shah is hardly comparable to "the Old Regime" in these United States, which you suggest "is the Constitution and the Declaration, free markets. etc."
Comparison is not to a set meaning of the word “conservative.” Conserve means to retain, save, etc. The Iranians would be conserving the Theocracy or more remotely the Monarchy. American conservatives would be conserving Constitutional Government, our founding and precariously continuing system. An Iranian advocating American style Constitutional government would be a total Radical. Liberal has a meaning with ideological components. “Conservative” has historical components relative to its society. It is not like a name. It is not “George” or “Phyllis” that would refer to one discrete system in one particular time and place.