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The child is a citizen, and on that basis the immigrants can get permanent residence status, to be in the US with their citizen child.

Oh really? As written, the 14th Amendment was NOT intended to grant citizenship to the children of foreign subjects.

The Slaughterhouse Cases are the first Supreme Court interpretation of the 14th Amendment on record. The author of the majority opinion is a contemporary of those who drafted and debated the Amendment. The following text is from the majority opinion (about 3/4 of the way down the linked source page):

http://www2.law.cornell.edu/cgi-bin/foliocgi.exe/historic/query=[group+f_slavery!3A]/doc/{@6621}/hit_headings/words=4

Slaughterhouse Cases, 83 U.S. 36 (1872) (USSC+)
Opinions
MILLER, J., Opinion of the Court

"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 first observation we have to make on this clause is that it puts at rest both the questions which we stated to have been the subject of differences of opinion. It declares that persons may be citizens of the United States without regard to their citizenship of a particular State, and it overturns the Dred Scott decision by making all persons born within the United States and subject to its jurisdiction citizens of the United States. That its main purpose was to establish the citizenship of the negro can admit of no doubt. The phrase, "subject to its jurisdiction" was intended to exclude from its operation children of ministers, consuls, and citizens or subjects of foreign States born within the United States.