On the one hand, it is hard not to overdose on the irony here: conservatives calling Kim Davis Rosa Parks, even as most of their ideological forbears never actually liked the real one. For right-wingers to hold up one of their own as equivalent to the soldiers of the civil rights movement would be hilarious were the claim not wrapped in such an earnest lack of self-reflection. After all, the conservative movement in the 1950s and 1960s detested Dr. King and those with whom he worked to break the back of American apartheid. And no, the votes of Republicans in the House and Senate for the Civil Rights Act doesn't change that. Virtually none of those Republicans would be considered conservative by today's right-wing cognitariat (just as none of the southern Democrats who opposed the law could be considered liberal). The conservative movement -- best represented by publications like The National Review, or activists like Phyllis Schlafly and candidates like Barry Goldwater -- made quite clear their opposition to the civil rights struggle. In the case of Buckley's publication, they went so far as to suggest that the bombing of the 16th Street Baptist Church in Birmingham---which killed four young girls and is one of the most iconic acts of white terrorism in the history of the nation---might well have been the act of a "crazed Negro," hoping to whip up sympathy for his otherwise unjust cause.
Because the right-wing played no part in the civil rights struggle, other than opposing it, and applauding when law-and-order sheriffs beat up peaceful demonstrators---or claiming that the Selma-to-Montgomery march was really just an excuse for black men and white women to engage in "sex orgies" on the side of the road---it makes sense that today's conservatives would utterly misunderstand the lessons of that movement. But for the sake of those who might be vulnerable to the revisionism of FOX and the modern Republican Party, let us review.
First, Dr. King and the civil rights movement were seeking to ensure the rights and liberties of those who had for so long been denied them. They were fighting for equal rights, not merely their interpretation of God's will. They were arguing that the denial of opportunities to people of color on the basis of their identity as people of color was an abrogation of the nation's promises of equality and a violation of the Constitution. Dr. King was a preacher, but his argument was not that segregation should be struck down because God said so; rather, he argued that segregation should be struck down because it was unjust in civil society and in contravention of the highest laws of the nation.
By contrast, Kim Davis and her supporters -- who believe same-sex couples should not be able to marry, or that those who oppose such marriages on Biblical grounds should be allowed to discriminate against them on that basis -- are not fighting to ensure equal rights. Quite the opposite. They are seeking to restrict marriage to heterosexuals only, thereby perpetuating unequal access to this opportunity. In that regard, rather than Rosa Parks in the Montgomery bus scenario, they are effectively the bus drivers. They are seeking to uphold discrimination and the special favoriting of heterosexuals, much as those bus drivers were upholding discrimination by sending blacks to the back of the bus, treating them as second-class citizens, and thereby favoring white riders. To argue that someone seeking to uphold unequal treatment is the moral equivalent of someone seeking to secure equal treatment is a perversion of language and history hardly worth serious consideration. And yet, such a perversion is utterly mainstream among American conservatives who have done just that by making Davis their martyr.
Second, to the extent that many segregationists (and most all slaveholders) defended their position on the basis of religious beliefs---ones to which they adhered every bit as sincerely as Davis with regard to same-sex marriage---to suggest that such beliefs should be allowed to trump the Constitutional mandate of equal treatment (as articulated by the courts) would again place Davis and her defenders in the camp of those prior bigots, not those who were challenging them. By the logic of those who suggest Davis should be allowed to keep her government position, even as she refuses to dispense the service of that position equitably, teachers with a religious objection to so-called race-mixing should be able to refuse to teach black children in integrated schools and still keep their jobs. County clerks with religious beliefs against interracial marriages should be able to deny such couples marriage licenses, much as Davis is doing for gay and lesbian couples, while still keeping their jobs. Voter registrars with religious beliefs in racial separatism or white supremacy should be allowed to refuse to register black voters, all while remaining on the public payroll.
Beyond racial examples, such deference to one's "sincerely held religious beliefs" would allow a number of other horrific abrogations of public responsibility as well. For instance, police who sincerely believe the Bible's repeated admonitions that disobedient children should be put to death could refuse to arrest parents who murder their back-sassing teenager. District Attorneys with the same "sincerely held" views could refuse to prosecute them because doing so would "violate their conscience" and their interpretation of God's will. Likewise, a law enforcement official who believed that a woman battered by her husband was insufficiently "submissive" (as demanded of her in multiple Bible verses) could refuse to arrest the abuser, or shirk their investigative responsibilities in keeping with their sick (but quite literal) interpretation of Scripture.
Third, the claim that Kim Davis has a right to her beliefs and religious convictions, while true, has no bearing on her rights as a county clerk. She is not entitled to her job, nor the ability to keep it while refusing to actually do it. Her paycheck is paid for by the taxpayers, including gay and lesbian taxpayers. To suggest that she should be able to draw on public money while refusing to treat all who make up that public equitably is to suggest that LGBT folks should be expected to pay for the performance of services to which they will receive no access---an obviously unfair arrangement, and a particularly disturbing form of taxation without representation. Those who believe as Davis does have every right to their beliefs, and to act on the basis of those in their jobs, so long as those jobs are with religious institutions, which by definition tend to be exclusive and which are exempted from the expectations of other non-discrimination laws. But if you have a public sector job, or one in the private sector but which involves the provision of goods and services to the general public (as with interstate commerce), then your right to discriminate is circumscribed. To hold otherwise would allow the re-segregation of lunch counters, restaurants, movie theaters, and more. Thus, for Davis's boosters to equate her actions with the folks who fought to end such discrimination, even as her position would enshrine the right of segregationists to return to it (so long as they couched their arguments in religious belief) is especially vile.
And no, it will not do at this point in the discussion to retreat to the old canard that "sexuality is different than race," presumably because race is immutable and sexuality is somehow "chosen." Putting aside the question of the fixity of sexual and affectional orientation (or for that matter the fact that race is not actually biological, as commonly believed), such an argument is completely irrelevant to the issue of Davis's religious beliefs. She is not claiming that same-sex marriage is wrong because homosexuality isn't biological, whereas race is. No doubt Davis would refuse to marry same-sex couples, because she thinks God said so, even if someone were to march right up to her with incontrovertible proof of a gay or lesbian gene. The issue is not whether discrimination on the basis of one thing or the other is more or less acceptable because of various scientific distinctions. Indeed, to pin one's religious notions on what science does or does not say would be inherently comical. No, the issue is, should one be able to discriminate in the provision of a public service on the basis of one's private religious beliefs, whether or not those beliefs are backed up by science. In that regard, and in evaluating the implicit hypothesis involved, religious beliefs about race, sex, gender, sexuality or any other matter are all equally valid as tests of that hypothesis. If one can be allowed to do one's job (or not) based on sincerely held religious beliefs about sexuality, then one would have to be allowed to do one's job (or not) on the basis of one's sincerely held religious beliefs about anything.
To insist that Kim Davis is heroic, let alone on a moral par with those who fought to end legalized racial discrimination in America is simply more evidence of how ignorant, politically craven, or both, the nation's conservative movement really is. They are not fighting for the preservation of the rights of Christians, which rights are well protected and endangered only in their fever dreams. Rather, they are fighting for the maintenance of Christian privilege and hegemony---for the right of Christian belief to subvert secular law. They must be defeated entirely in their quest for it.