Could be that the birth certificate shows that his parents were never actually married. Don't know why that would embarrass him particularly (there are no illegitimate children, just illegitimate parents) but he may have feared it would cost him some votes.
I think his birth certficate says Barry Dunham parents unwed. He’s spent so much time covering it up, beginning back in the days when being born out of wedlock would have mattered, and also in an attempt to fix his own story (who wants to accept the fact your father abandoned you?) that he can’t come clean now. His books, his whole “story” is based on the lies he told in the books he wrote. Coming clean now, even on this which is a nonissue for so many in our culture, would lead to other questions about his integrity. What else did he lie about?
Oh what a tangled web we weave when first we practice to deceive. One lie leads to another and then another and so and so on....