Actually Parker never exposed him in the comics. In the comics Spidey happened to capture the real serial killer the Sin-Eater after he kiled Captain Jean DeWoulff (Spider-Man's friend and possible love interest).
But apparently Brock working at a rival paper (you never meet him until after he is Venom) wrote an article claiming that his source was the Sin-Eater and exposed him. After the real serial killer was caught, it turned out Brock didn't fact check his source who was a compulsive liar and confessor and thus Brock's career was ruined and instead of blaming himself or even the guy who pretended to be the Sin-Eater or the Sin-Eater himself, for some reason he weakly chose to blame Spider-Man for capturing the real killer.
Yeah...I keep telling people Brock's origin in the comics sucks. Even if you don't like the dooplenganger angle and wish Venom was more of anti-hero/Robin Hood type like in the comics (something I didn't), his origin sucks.
Anyway, I think under these circumstances without the symbiote Peter would have gone straight to J. Jonah Jameson and shown him the evidence and gotten Brock justly fired. He would not have tried to pick a fight though in the news room and rub Brock's face in it after shoving him into a picture frame and taunting him first, however.