They did break the law

...both of the parents did, actually. Passport fraud; entering the country on some one else's passports. But their daughter Noriko has known no other life than that of a child in Japan, and is now a 13 year old student in public Junior High School (something that most children of legal migrants to Japan often do not achieve; they get stuck fulfilling legal public education requirements by attending privately run schools that often teach in the foreign language of the migrant community... not so good for social integration, that).

But the ruling is that only the daughter may remain in Japan as a resident. The parents *have to pay the price of breaking the law*. But after that, there are options available.

The justice ministry, meanwhile, revealed Friday that it intended to grant Noriko, who only speaks Japanese and attends a junior high school in Warabi, Saitama Prefecture, special permission for residence later this month so she can continue to go to school in Japan.
...
Earlier this month, Justice Minister Eisuke Mori said that Japan would be willing to grant short-term reentry permission to the girl's parents after they leave the country.

In the event of deportation, the parents would normally not be allowed to revisit Japan for five years due to the record of their illegal stay in the country, while in the case of voluntary departure they would be barred for one year.

But the justice minister suggested he would allow them to make short-term visits.

Keeping families together is a basically decent thing to do.

Finding a way to do so would be a credit to Japan, and offer some hope of resolving the issue of assimilating the significant number of migrants of questionable or illegal status that are a fact of life here.

Having the houmusho (Ministry of Justice, Japan) find a way to offer a role to Noriko Calderon and specifically to her parents Arlan and Sarah to make a public campaign to bring in to legal compliance visa-overstayers and false-entry residents, and to make a scheme that allows such individuals to convert their status to the legally correct form without having to leave Japan would be a good first step...

...and a way for the Calderon family to show some appreciation for the forgiveness being offered.

Japan, like any country, should welcome those who learn to play by the rules.

Even if that knowledge comes late.