Flight instructor reported while in the pattern another aircraft was not allowing for spacing resulting in the flight instructor taking evasive action to avoid a collision with the other aircraft.

Narrative

I was performing commercial training with a student. There were two other aircraft in the pattern; one on final and another on upwind turning crosswind. We made an announcement 7 miles out and 3 miles out on our intentions. We were going to enter downwind and we made that call until Aircraft Y decided to not respond to us. We then announced that we would follow behind them on the downwind. As we were setting up for the short field landing they followed very close behind us but we were still able to complete the landing. However; I told the other aircraft to maintain better separation. They just said they saw us. When my student completed his short field landing we then took off again. They did a go-around as were turning crosswind. As we performed our gear down go down on downwind operations Aircraft Y was already on the downwind. We announced that we were turning base and final. Aircraft Y insisted on doing a simulated engine out. I told them we are on a short final do you see us? She insisted she did but did not extend; instead; she just headed right towards us. We went around immediately to avoid them and they had to go around as well as they couldn't make the runway. They flew parallel with us and came very close to us. This is unacceptable behavior and this is uncalled for.