A tweet from the account of Jens Maier, a former judge, had attacked Noah Becker for reportedly complaining about being seen as the "eternal son" of his famous father.
"It seems the little half-negro simply got too little attention," read the tweet posted from the account of Maier, one of more than 90 AfD members elected to parliament last September.
Maier claims he did not write the tweet, which has since been deleted, saying one of his staff had posted it.
In a statement issued Monday, Maier apologized to Becker for the "lapse" by his employee, who he said no longer worked for him.
"This tweet not only contradicts my style but also does not reflect my ideas. I will ensure that this does not happen again," he said.
Maier's explanation appeared to have been accepted by the party's leadership, which warned him to take greater care in managing his employees.
Noah Becker has filed a criminal complaint against Maier over the tweet, which was also blasted by his father in a scathing column in Sunday's edition of Die Welt newspaper.
"Jens Maier says such things neither out of stupidity nor fear. He knows exactly what he is doing and why," wrote Boris Becker, demanding that the MP face "consequences".
It was a second time in a week that AfD deputies had come under fire after police filed a complaint against senior party member Beatrix von Storch over a New Year's Eve tweet which they say violated laws against incitement to hate.
Von Storch had criticized Cologne police for sending a New Year's greeting in Arabic on Twitter, asking if authorities had meant "to placate the barbaric, Muslim, gang-raping hordes of men?"
On January 1st, a law against online hate speech went into effect in Germany, requiring social media companies to remove illegal inflammatory comments or face up to €50 million ($60 million) in fines.
The AfD capitalized on discontent against a mass influx of asylum seekers to Germany since 2015 to make the strongest showing for a far-right party in a national election in the post-war era.