Are parents of Muggle-borns subject to the Statute of Secrecy?

Question

A lot is made of the Statue of Secrecy in the books. However, what about the Muggle parents of witches or wizards? Do they have to sign like an official secrets act? Hundreds of non-wizards knowing about the wizarding world has to be a hazard?

Answer 1

They aren’t bound by the Statute, but do have to keep the secret.

There’s a bit more detail that can be inferred on how the Statute of Secrecy applies to Muggle families of wizards in the Pottermore writing by J.K. Rowling on Professor McGonagall. Isobel McGonagall explains that she and Minerva are bound by the Statute of Secrecy. However, she doesn’t say that Robert, her Muggle husband, is also bound by it, so it seems that he’s not. Also, after she tells him, he doesn’t seem to be made to sign or swear anything in front of the Ministry where he promises to keep magic a secret. He still seems to be expected to, though.

Although Robert McGonagall loved his wife no less upon discovering that she was a witch, he was profoundly shocked by her revelation, and by the fact that she had kept such a secret from him for so long. What was more, he, who prided himself on being an upright and honest man, was now drawn into a life of secrecy that was quite foreign to his nature. Isobel explained, through her sobs, that she (and their daughter) were bound by the International Statute of Secrecy, and that they must conceal the truth about themselves, or face the fury of the Ministry of Magic.
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It seems likely that the Ministry of Magic doesn’t have jurisdiction over Muggles and can’t hold them accountable to wizard laws. However, they still wouldn’t want Muggle family members of wizards trying to reveal magic, but it seems they have to control that in other ways. The Muggles obviously wouldn’t be able to demonstrate magic in an attempt to reveal it, and presumably wouldn’t want to do something that could harm their families. However, as family members of wizards, these particular Muggles would have access to magical objects and could cause trouble if they show actual evidence to other Muggles in a way that they’ll believe it. If something like that happens, it’s likely the people they tell would be Obliviated, and they’d get some sort of warning. It’s highly unlikely that the Ministry would be unprepared should such a situation arise.

Most “reveals” should be low-level and relatively harmless.

It’d also help that, largely, Muggles are unwilling to simply believe in magic without proof. If the Muggle family member claims magic exists without showing some sort of proof, like they get drunk and start telling people at the bar how their child just learned to turn hedgehogs into pincushions, they’d likely be dismissed as a lunatic and the wizarding world would remain a secret.

“If any Muggle is unwise enough to confide in another that he has spotted a Hippogriff winging its way north, he is generally believed to be drunk or a “loony.” Unfair though this may seem on the Muggle in question, it is nevertheless preferable to being burnt at the stake or drowned in the village duckpond.”
_{^{- Fantastic Beasts and Where to Find Them}}

This is confirmed by another J.K. Rowling writing on Pottermore explains why Muggles who marry wizards haven’t yet exposed the existence of magic to the larger Muggle world. The same thing would apply to Muggles who have wizard children.

Over subsequent decades and centuries, the number of mixed marriages began to climb again until the healthy levels of today, and this has not led to widespread discovery of the hidden magical community. Professor Mordicus Egg, author of The Philosophy of the Mundane: Why the Muggles Prefer Not to Know, points out that Muggles in love generally do not betray their husbands or wives, and Muggles who fall out of love are jeered at by their own community when they assert that their estranged partner is a witch or wizard.
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Most cases are harmless, any ‘more dangerous’ ones would be outliers. However, it’s almost certain that the Ministry would step in and handle any cases where Muggles start believing.

Answer 2

I think that wizards assume that laws about Muggles telling others would be largely unnecessary, since Muggles would not believe other Muggles telling them about magic, and the Muggles who know would be afraid to talk about it for fear of being considered mad. This is in fact borne out by Fudge's conversation with the prime minister in the beginning of Half-Blood Prince (my emphasis):

“But,” said the Prime Minister breathlessly, watching his teacup chewing on the corner of his next speech, “but why — why has nobody told me — ?”

“The Minister of Magic only reveals him- or herself to the Muggle Prime Minister of the day,” said Fudge, poking his wand back inside his jacket. “We find it the best way to maintain secrecy.”

“But then,” bleated the Prime Minister, “why hasn’t a former Prime Minister warned me — ?”

At this, Fudge had actually laughed.

“My dear Prime Minister, are you ever going to tell anybody?”

Still chortling, Fudge had thrown some powder into the fireplace, stepped into the emerald flames, and vanished with a whooshing sound. The Prime Minister had stood there, quite motionless, and realized that he would never, as long as he lived, dare mention this encounter to a living soul, for who in the wide world would believe him?