The bowler was fined 15% of his match fee and given a demerit point for his behaviour on Saturday as England went on to clinch the series opener in Manchester by three wickets.
After England's 'Barmy Army' supporters group tweeted the news of his sanction on Tuesday, Broad said of his father: "He's off the Christmas card & present list."
Chris Broad would not normally officiate in a match involving his son but he is overseeing all six of England's home tests this summer because of travel restrictions imposed to combat the spread of the novel coronavirus.
It was the bowler's third demerit point in the last 24 months. Another demerit point picked up in the second test, which begins in Southampton on Thursday, would trigger a one-test suspension.