The detective at Parland was erroneous. Not, "could this"? There was no dead detective.
Was Parkland close to the Brewer location? No.
Was Tippit taken to Parkland? No.
Are you grasping at straws?
Not known to be erroneous. Not 100% correct? Definitely.
It could be a totally erroneous report. Like that of the Secret Service agent killed at Dealey Plaza. Or it could be an early report about the shooting of Officer Tippit, with someone mishearing some of the information they were getting over the police radio, while getting some details right.
At the about that time, around 1:25, the police radio was talking about the need to get blood to Parkland. I would assume for Governor Connally. The mention of Parkland could have confused someone.
In any case, a totally erroneous report, or a valid report with significant errors, this could be the basis of Mr. Brewer remembering hearing about the shooting of a Dallas police officer just before Oswald showed up.