However, a head clash during the first round left him with a deep gash above his eye.
Parker received 14 stitches on the cut, including five under the skin.
"We are erring on the side of caution and have decided we will not spar again until next year," Barry said.
"I have seen [similar injuries] ruin fighters' careers by being handled the wrong way.
"That is why we are taking as much time as he can. We are not rushing into a fight."
Parker's next fight will be against Brazilian heavyweight champ George Arias (45-11) in March 2014.
The fight was originally scheduled for next month, but Parker's injury has forced the fight back four months.
"We are not making this a negative. For us this is more like a positive," Barry said.
He said they sparred more than 130 rounds in the lead up to the fight against Tatupu, but for the next few months it would be all about strength, endurance, resistance and weight training.
"We will use this time to put a bit more muscle on him."
The hard work will suit Parker who said he simply loves to train.
"I love training in Vegas. I know the importance of keeping fit and keeping in great shape," Parker said of his immediate plans.
"It is easy as to put on the weight but it is harder to keep in great shape."
He said he had learned a lot from his last fight, specially after dealing with the pressure of almost having the fight stopped because of his injury.
"After the first round, in the break, the doc was, like, you have got one round left. It snapped me back into the game plan. It was a positive being able to snap back and go back to the plan."