My favorite example among major Presidential candidates was John Charles Fremont, the first Republican candidate for President. Not only was his father French, but his father's nationality was *very* well-known.
Another good recent example is Ralph Nader. His parents were known to be immigrants, and his campaign had a huge impact on the 2000 election. If eligibility always required two citizen parents, it's curious that *no one* ever asked Nader for proof of his parents' citizenship, considering that there would have been such a huge benefit to one party if he'd been disqualified.
Facts are stubborn things; and whatever may be our wishes, our inclinations, or the dictates of our passion, they cannot alter the state of facts and evidence -John Adams