Pitman smashed home a 40th-minute free-kick from 25 yards with Crawley's defensive wall in apparent disarray.
But Barker felt Premier League referee Lee Probert should not have allowed the goal to stand.
Barker said: "I did not think he should have been allowed to take the free-kick that quickly. But we cannot use that as an excuse for losing as heavily as this."
Crawley were struggling from the 16th minute when their striker Gary Alexander headed Steve Cook's cross past his own goalkeeper.
Pitman's goal came after Marc Pugh had been fouled on the edge of the area and there was no way back for Crawley once Probert let it stand.
Barker's team were much-improved after the break and only a crucial save with his legs by Bournemouth goalkeeper David James denied substitute Billy Clarke a 70th-minute goal.
Bournemouth made sure of the points in the 84th minute when Simon Francis and Lewis Grabban set up a chance for midfielder Eunan O'Kane to run clear and score.
Barker said: "There were a few players out there who did not work hard enough. One or two did not step up to the mark, maybe it was because of tiredness. Bournemouth are a good side and with the funds they have available you would expect them to be.
"But in praising the opposition I have to say that we were not at our best and we made it easier for them than we should have done. We can blame all kinds of things, but in the end we have been beaten by three goals and that says it all."