Democrats were gleeful when Republicans Chris Christie and Rand Paul got tangled this week in the debate over the Disneyland measles outbreak, until similar words from the pasts of President Obama and 2016 front-runner Hillary Rodham Clinton came out.

Mr. Christie walked back his recent comments calling for “balance” and a role for “parental choice” on childhood vaccines, saying in a Monday statement that “with a disease like measles, there’s no question kids should be vaccinated.”

But the Democratic posturing as the party of science was short-lived. Several websites promptly dug up a comment from Mr. Obama during the 2008 presidential campaign in which he said the research on whether vaccines cause autism was “inconclusive.”

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Then it was reported Tuesday that Mrs. Clinton had responded to a questionnaire during the 2008 presidential primary race from a vaccine skeptics group in which she said, “I am committed to make investments to find the causes of autism, including possible environmental causes like vaccines.”

The episode reveals that the emotionally charged vaccination issue has political pitfalls for both parties as candidates line up for the 2016 presidential contest, even though a Pew Research Center report released last week shows Democrats and Republicans alike strongly support childhood vaccinations.

A left-right coalition

For Republicans, the trick lies with soothing the party’s libertarian wing, which reflexively resists government intervention in the decision-making of parents. For Democrats, the challenge is figuring out how to put Republicans on the ropes without alienating the large pockets of “anti-vaxxers” in heavily Democratic areas such as Los Angeles and San Francisco.

“This is one of those cases where the far left and the far right converge,” said Republican political strategist Dick Wadhams. “I just think it’s poor judgment for anyone to suggest that children should not be vaccinated. It defies logic that someone could even suggest that in this day and age.”

Democrats jumped on the issue this week following comments by Mr. Christie and Mr. Paul, Kentucky Republican, who said Monday that vaccines “ought to be voluntary” during a radio interview with host Laura Ingraham.

“I’m not arguing vaccines are a bad idea. I think they are a good thing, but I think the parent should have some input,” said Mr. Paul, an ophthalmologist who added that he did have his children vaccinated. “The state doesn’t own your children. Parents own the children.”

Mr. Christie and Mr. Paul are both oft-mentioned potential candidates for the 2016 GOP nomination, making them juicy targets for the Democratic National Committee, which released a statement Monday blasting the Republicans as anti-science.

“Republican after Republican is bowing to the rhetoric of the anti-vaccination movement instead of standing up for the science supported by almost all doctors and scientists on protecting our kids and keeping our nation safe,” the Democratic National Committee said in a Monday statement.

Mrs. Clinton chimed in with a Tuesday post on Twitter saying, “The science is clear: The earth is round, the sky is blue, and #vaccineswork,” she tweeted. “Let’s protect all our kids. #GrandmothersKnowBest.”

Democratic resistance

But Democrats have their own cultural vulnerabilities on the issue. Among the communities hit hardest by the recent measles outbreak are Los Angeles and the Bay Area, where Democrats vastly outnumber Republicans and where the anti-vaccination movement is on the rise.

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