My father, Charles F. Palmer, a law school classmate of Bill Clinton, took the photo.
I have a photo of the same exact place and time, with Bill and Hillary in same clothes, but with my mother in the photo. Bill had given them a ride to the train station. Just uploaded it to my twitter feed: @oatespalmer.
My mother and father came to Yale in the fall of 1970, having dated and lived together in Berkeley and in Washington DC. My father was in Bill Clinton's law class, the class of 1973. (Bob Reich, the future Secretary of Labor, was also in their year.) My mother, Marylouise Oates, received her Masters of Divinity from Yale around the same time. Hillary Rodham was in the Yale law class of 1972, even though she was younger than both Bill Clinton and my father - she went straight to Yale after she graduated from Wellesley.
Shortly after Bill Clinton was elected President, my mother found this photograph (below) in a photo album somewhere and gave me a copy. When Hillary Clinton wrote her memoir, Living History, I saw that the photograph of just her and Bill from Yale was on the back cover -- and was credited only to "courtesy of author's private collection" or something like that. I recognized it as likely being taken around the same time as the photo with my mom, and thought it could have been taken by my dad, too, but didn't really explore it.
Probably about a year ago, people were posting the "just Bill and Hillary" Yale photo on Facebook, and I looked closely at the two photos side by side and saw that they were clearly taken at the same time and the same place -- same building behind them, and Bill Clinton in virtually the exact same pose. I posted the two photographs on my Facebook page.
A few months later -- just the other day -- a friend who is friends with a Quora staffer sent me the link to this question. I have always wanted to see my dad, now a Superior Court Judge here in Los Angeles, given credit for taking this iconic photograph of the two Clintons.
I have the photo of the Clintons and my mom framed in my house. When I show someone the photo, I like to let them figure out who is in the picture. They inevitably have one of two responses: "HOLY $#&$ LOOK AT BILL CLINTON'S HAIR" or they ignore the Clintons completely -- not recognizing them -- and instead say, "Check out the boots your mom is rocking!"