TOUCHDOWN BROWNS! pic.twitter.com/RjRt9DVlpB— Cleveland Browns (@Browns) October 16, 2016
The league's social media policy, which went into effect Wednesday, forbids teams from posting highlights from television directly to social media or turning highlights into GIFs.
When asked about the parody highlights, the NFL said it would have no comment.
NFL Players Association spokesman George Atallah noticed the tweets and took the opportunity to tweet his own shot at the NFL.
The social media policy also forbids teams from shooting video inside the stadium during the game and posting it on social media. It also forbids using Facebook Live or Periscope to stream anything live within the stadium.
Two high-ranking team executives told ESPN on Sunday that, given the reaction of many teams, the new social media policy is expected to take up a greater part of the agenda at this week's owners meetings than originally expected.
ESPN's Darren Rovell contributed to this report.