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It is interesting to note the case Perkins v. Elg, 307 U.S. 325 (1939), which refers to a young woman born in the US of parents holding Swedish citizenship. Her father had been naturalized, but later renounced his US citizenship after the parents returned to Sweden with her when she was a child. She sued to claim her US citizenship on reaching age 21:

The court below, properly recognizing the existence of an actual controversy with the defendants..., declared Miss Elg 'to be a natural born citizen of the United States'

By this definition, therefore, anyone born in the US, even of non-citizen parents, is a natural born citizen of the United States.

The website's "Natural Born Citizen Chart" claims the the Constitutional definition of "Natural Born Citizen" requires that the person be "Born in the U.S. mainland" of parents who "Both are U. S. Citizens". This is not true. The term is not defined in the Constitution: "No person except a natural born citizen, or a citizen of the United States, at the time of the adoption of this Constitution, shall be eligible to the office of President..." The site claims that this term is also defined in law, but offers only a Swiss source document predating the Revolution:

Emmerich de Vattel's explanation of "Natural Born Citizen" given in his 1759 benchmark work, used, and so often quoted, by the framers of the U. S. Constitution, makes the understanding simple, explicit, clear, definite, exact, precise, and strict. In the CITIZENS AND NATIONS, paragraph #212, de Vattel says: "The natives, or natural-born citizens, are those born in the country, of parents who are citizens. As the society can not exist and perpetuate itself otherwise than by the children of the citizens, those children naturally follow the condition of their fathers, and succeed to all their rights. The society is supposed to desire this, in consequence of what it owes to its own preservation; and it is presumed, as a matter of course, that each citizen, on entering into society, reserves to his children the right of becoming members of it. THE COUNTRY OF THE FATHERS IS THEREFORE THAT OF THE CHILDREN."

This statement is obviously in conflict with the concept of the US as a nation of immigrants, and with US law, specifically the 14th Amendment:

All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the state wherein they reside.

The issue of his father's citizenship is therefore irrelevant, and the only issue in question is whether Obama was, in fact, born in the US.