"Get a grip little boy," she said.
"When you grow up you will realise that life is not black and white.
"If you ever have the privilege of serving your country you will find it is not easy and difficult choices have to be made.
"Until then try not to be such a judgmental little c**k," Ms Cooper said.
This afternoon, Ms Cooper apologised on Facebook for her comments.
"I'm very sorry if my comments to Daniel O'Connor on the Auckland Pride Festival FB page have caused offence. On reflection, my choice of words to make a point was unfortunate and I apologise if my actions have detracted from what was otherwise a fantastic event of celebration and pride."
A council spokewoman said it had received one code of conduct complaint about Ms Cooper's comments on Facebook in a conversation around "MPs" attendance at the pride parade.
"This will be looked into by the chief executive and managed in the usual way," she said.
The code of conduct for elected members says councillors should not make derogatory terms towards others, including in the media, and treat people with courtesy.
If the complaint is upheld, the most likely penalty would be censure for Ms Cooper.
Mr Brown, who was censured by the council over his affair with a council adviser, declined to comment on Ms Cooper's outburst.
