First-half goals from Demarai Gray and Abdoulaye Doucoure had put the Toffees on course for a third win in six games under Dyche at the City Ground.
But Brennan Johnson, who cancelled out Gray’s early penalty, ensured his side earned a share of the spoils with his second 13 minutes from time and kept Everton, and the other sides below them in the relegation battle, at arm’s length.
But Forest had issue with the build-up to Doucoure’s goal as there appeared to be a foul on Morgan Gibbs-White that Brooks ignored, while Dyche was “amazed” his side did not get a second penalty after Seamus Coleman went down under a challenge from Jack Colback.
Cooper said the official had “too much impact” on the game.
“There will be a lot of talk about the referee’s performance. Both teams were going for it but the referee has had too much involvement really,” he said.
“I’m disappointed, I’m not going to say anything that is going to get me in trouble but he had too much impact on the game and there were some defining moments.
“The key one was the decision that ended up with the ball going in the box for the second goal. Yes we should do better with it but it shouldn’t have gone into our box because it should have been a foul the other way to Morgan.
“And when (James) Tarkowski gets booked on (Renan) Lodi, fine, yellow card, no problem, but he has blown up and Brennan has a one-v-one.
“That cannot happen at this level. It’s not good for him either, because he is going to be gutted with that and probably will be looked at, I hope, by the people responsible.
“It’s a pity things like that are affecting games.
“They have done alright out of the referee today, Everton. And to be fair there were some small things that went our way, but the big things went for them.
“It was two teams having a go and both teams will want to win or lose the game by what we do as a team, and, unfortunately, I’m not saying it’s all down to the ref, but they are just moments that can’t happen at this level.”
Dyche said of the first-half incident involving Coleman and Colback: “Amazed we didn’t get a second penalty, but I don’t know what’s a penalty any more.
“You touch someone on the shoulder and they go down, it’s a penalty. Seamus gets there, puts his foot in front of him, he kicks his foot and it’s not a penalty. I don’t know where that lives.
“I asked the referee and he didn’t think it was a penalty. I said ‘I don’t know what is then’. My way of looking at it is, if someone has broken in front of you, if you touch or are around them, even in my day, it is a penalty.”
The draw kept Forest four points above Everton in the relegation battle, but the Toffees appear to be on an upward trend under Dyche, having taken seven points from his six games in charge.
Dyche added: “Overall it was a very strong performance on the back of a big shift at Arsenal. I just see it as growth.
“We have to keep getting points on the board but you have to have a method to what you are doing and there are very pleasing signs to what the players are delivering and how they are taking these games on.
“We need points of course, but there has to be a plan, a method and reason of getting them.”