This is the full answer about "dual national" from the VWP Improvement FAQ cited by another answer:
How is “dual citizen” or “dual national” defined? What if I was born in a country, but never lived there and do not consider myself a
national or citizen?
We will make nationality determinations in accordance with U.S. legal
standards and practices, not merely by reference to the laws and
practices of foreign governments. If an individual believes that he
or she is eligible for an ESTA travel authorization, the individual
should apply for an ESTA, answer all questions truthfully and
accurately, and that individual’s eligibility for an ESTA
authorization will be determined in accordance with U.S. law. If you
have any questions, please contact CBP at 1-202-344-3710.
I agree that "in accordance with U.S. legal standards and practices" is not very specific, but we aren't entirely in the dark since it is U.S. legal standards and practices that also determine U.S. citizenship practices.
I've modified from the original to make the argument less contentious, though I believe it is the same argument. Under US law citizenship is voluntary and there is a fundamental right to expatriation if you might have it but don't want it. The original expatriation law (R.S. § 1999) was passed in 1868 at the same time automatic jus soli citizenship was added to the constitution, and says this:
Whereas the right of expatriation is a natural and inherent right of
all people, indispensable to the enjoyment of the rights of life,
liberty, and the pursuit of happiness; [...] Therefore any
declaration, instruction, opinion, order, or decision of any officer
of the United States which denies, restricts, impairs, or questions
the right of expatriation, is declared inconsistent with the
fundamental principles of the Republic.
What this means, as is made clearer by the context in which it was enacted, is that even if you have an automatic obligation of or right to citizenship (anywhere) you are free not to accept it or to abandon it, and be treated consistent with that choice under US law. What your past nationalities might have been aren't relevant. Current law (8 USC 1481), to which the above is attached as a note, allows US nationality to be shed by the mere intention to do so coupled with a foreign naturalization or formal oath of allegiance (i.e. there's an inhibition against abandonment to statelessness) if you are an adult (i.e. your parents can't make an irrevocable decision for you).
Given this, my best guess is that the US would never consider someone who is a national of somewhere other than Iran, who claims no Iranian nationality and who has taken no action (travel, obtain documents, join the army) that would suggest otherwise, to be an Iranian national no matter what the parents' nationalities were and no matter what Iranian law says. It also isn't a coincidence that the ESTA nationality question is present tense, to be consistent with the law above. If they were interested in possible past connections with Iran they'd also ask about places one has lived and/or the birth places of the parents, but since they didn't ask that there's no reason to answer that. That's all just a guess, though.
I hence think the best answer is that the OP should call 1-202-344-3710, explain his situation and ask them what they think.
P.S. So I phoned that number myself and presented the person that answered with the following entirely-made-up facts about a person:
- Born in Hong Kong, has Hong Kong and UK nationality
- Has a valid ESTA listing those nationalities
- Father is an Iranian national
- Birth not registered with Iran
- Has relatives in Iran, but has never been there
I then asked if the person now required a visa. The lady I was talking to asked two things:
- Where was the person born? I said Hong Kong.
- Has the person ever had an Iranian passport? I said no.
The lady then said that no visa was required and the person could travel on her existing ESTA.
The actual facts for the OP might not be those but I assume if they are similar the answer will be the same if she phones (she may also talk to the same person, as the call sounded a whole lot more like a not-very-busy phone on someone's desk than it did a call centre operation). This clearly doesn't preclude applying for a visa instead, but I think doing so for no purpose other than being able to mention the Iranian father because the ESTA application didn't ask about that might be committing a TMI error.