A critical point overlooked by the other answers is the credibility of the primary source.
It is a magazine article based on a few quotes by a single lawyer (Eva Becker) edited by an author "vhe".
The article provides no references to any court decisions or laws. That is somewhat uncommon: lawyers love to cite court decisions in blogs and articles. It may be intended for easier reading.
Despite the catchy title, the article is written towards parents who fear bad influence of ex-partners. The last paragraph titled "Wann sollten Sie einen Anwalt einschalten?" / "When should you consult a lawyer?" ends with "Hier finden Sie einen Anwalt oder eine Anwältin für Familienrecht in Ihrer Nähe." / "Here you can find a lawyer near you specialized for family law.". The entire article could be considered an advertisement.
No, the primary source cannot be considered a reliable source.
Further, compare the primary source with what the secondary source claims:
Wer „fremdenfeindliche“ Aussagen auf Facebook tätigt oder gegen
„Flüchtlingsheime“ demonstriert, könnte demnächst sein Umgangsrecht
mit seinem Kind verlieren, auch wenn keine Straftat vorliegt, erklärt
der Deutsche Anwaltverein.
The translation is almost good, except that I would not use an explicit "anyone", that would imply a "Jeder der ..." in the German:
Someone who makes "xenophobic" statements on Facebook or demonstrates
against "refugee homes" could soon lose his right of access with his
child, even if no criminal offense exists, the German lawyer's
association.
The primary source discusses only extreme examples (emphasis mine):
Äußert ein Vater oder eine Mutter hingegen vor dem Kind, dass man alle Flüchtlinge am besten umbringen würde, überschreitet er damit deutlich die kritische Grenze.
Translation (emphasis mine):
However, if a father or mother expresses in front of the Child, that all refugees should best be killed, a critical line is clearly crossed.
No, '"Xenophobic" statements on Facebook' are not in front of the child. Making said extreme statement on Facebook would however be a criminal offense (Volksverhetzung). Further it only says that a critical line is clearly crossed, it does not say that this directly leads to loss of access to the child.
Edit: The secondary source cites the primary one as follows, (emphasis mine on the difference)
Spreche ein Vater oder eine Mutter hingegen vor dem Kind offen eine Drohung gegen Flüchtlinge aus, überschreite das Elternteil damit deutlich die kritische Grenze.
(traslation)
However, if a father or mother expresses an open threat against refugees in front of the Child**, a critical line is clearly crossed.
The secondary source changed a "kill all" to "an open thread".
There is no indication, that the primary source changed their wording (web.archive from September 2015).
Regarding demonstration, the primary source says:
Schlimmer als Äußerungen wiegen meist Handlungen: Es ist etwas anderes, im Beisein des Kindes mit Bekannten abfällig über Asylbewerber zu sprechen, als das Kind zu fremdenfeindlichen Demonstrationen mitzunehmen. Doch auch hier zählt nicht Aktion an sich, sondern der Einfluss auf das Kind. Eine Mutter, die den Vorträgen eines sogenannten Hasspredigers beiwohnt, während das Kind in der Kita ist, und in der Zeit mit dem Kind über den Inhalt dieser Predigten kein Wort verliert, gefährdet damit noch nicht das Kindeswohl.
(translation)
Worse than utterances are usually actions: It is something else to to talk disparagingly about asylum seekers with acquaintances in the presence of the child, as to take the child to xenophobic demonstrations. But here, too, action does not count, but the influence on the child. A mother who attends the lectures of a so-called hate preacher, while the child is in the day-care center, and does not lose a word about the contents of these sermons during the time, does not endanger childhood.
No, the primary source does not claim that demonstrations can lose you access to the child.
Regarding the criminal offense, the primary source does explain that
Strafbarkeit zunächst nicht entscheidend
(translation)
Criminality is not decisive at first
But gives only examples that are not political (drug dealer, instigating the child against other family members).
The primary source does not support what the secondary source claims.
Note, the secondary source does mention one law: Paragraf 1684 Absatz 2 BGB. However, that just generally refers to the well being of the child (Wohl des Kindes).