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The peace movement (folk music, et al) of the 1960s had its focus, generally speaking, on two primary issues: the war in Vietnam and the civil rights movement.
Since the word "conservative" was used to describe what the movement was against, even though it "may not" have been used in a political context, but since it is now, most generally, associated with the Republican party, some historical perspective may be in order.
Civil rights:
The Civil Rights Act of 1957: This law, the first civil rights legislation passed in 80 years, was introduced by the Republican Eisenhower administration in 1956. It created a Civil Rights Commission and a civil rights division in the Department of Justice. It prohibited interference in the exercise of voting rights, and was an important first step which helped pave the way for additional legislation related to civil rights.
Civil Rights Act of 1964: Passed with the very strong support of Republicans and moderate support of northern Democrats, in both the House and the Senate, despite very strong opposition from House of Representatives southern Democrats who voted overwhelmingly against the bill, 92 to 11, and a filibuster in the Senate led by southern Democrats. Only one Republican senator participated in the filibuster against the bill, while 18 Democrats led the charge against the civil rights act. Twenty-one Democrat Senators voted against the Civil rights Act of 1964, while only six Senate Republicans opposed the bill.
Republicans, in general, were in favor of strong civil rights legislation, while Democrats, in general, especially southern Democrats, were strongly opposed. In fact, since 1933, Republicans had a more positive record on civil rights than the Democrats. In the twenty-six major civil rights votes from 1933 to 1964, a majority of Democrats opposed civil rights legislation in over 80% of the votes. By contrast, the Republican majority favored civil rights in over 96% of the votes.
Some of the Democrat Senators who voted AGAINST the Civil rights Act of 1964: Senator Al Gore, Sr. of Tennessee (Yes, his father), Senator William Fulbright of Arkansas (one of President Bill Clinton's political mentors), and Senator Robert Byrd of West Virginia (also a Ku Klux Klan member, and who has voted against all Supreme Court justice nominees who happened to be black, including the very liberal Thurgood Marshall). And let's not forget about Democrat Gov. George C. Wallace of Alabama, synonymous with racists and segregationists.
The Republicans don't always get the credit they deserve when it comes to civil rights. Civil rights activist Andrew Young (a Democrat mayor of Atlanta), wrote in his autobiography, An Easy Burden, that "The southern segregationists were all Democrats, and it was black Republicans who could effectively influence the appointment of federal judges in the South." Mr. Young admitted that the best civil rights judges were Republicans appointed by President Dwight Eisenhower, and were "among the many unsung heroes of the civil rights movement." By Andrew Young's own account, the elder Al Gore was a segregationist and not a champion for civil rights.
Vietnam:
In the late 1950s, President Eisenhower, a Republican, after consulting with U.S. advisors he had sent to Vietnam to assess the situation, advises that the French that they are involved in a hopelessly losing war in Indochina. He declines any significant involvement.
In 1961, President Kennedy, a Democrat, sent the first wave of 100 special forces troops to Vietnam.
Vice President Johnson, another Democrat, while personally touring South Vietnam in 1961, assures the South Vietnamese government that it is crucial to US objectives in Vietnam, and vows American support. In 1964, President Johnson turned down an offer for peace talks proposed by North Vietnam, and in 1965 he (Johnson) authorized the use of American combat troops. President Johnson oversaw the huge military buildup in Vietnam, going from 100 in 1961 to a peak of 550,000 in 1969.
In 1968, LBJ decided to not seek a second term. In 1969, President Nixon, a Republican, seeking an end to Johnson's Vietnam debacle, began to withdraw troops and initiated peace talks. In 1969 President Nixon signed a bill repealing the Gulf of Tonkin resolution, a resolution signed into policy by President Johnson 5 years prior.
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Conclusion:
The protestors of the "peace movement" of the 1960s weren't protesting any "conservative" position in the political sense. To the contrary, they were a very vocal movement against the policies of the current day Democrat party.
Very interesting, if you think about it.