Piastri produced the fastest lap of the two practice sessions for the season-opening Australian grand prix on Friday, overcoming power issues at the start of the day.
His flying lap of 1:19.729 at Albert Park put him 0.214 seconds ahead of Kimi Antonelli in the second practice session, while his Mercedes teammate George Russell was third.
Ferrari also had a positive day with Charles Leclerc and Lewis Hamilton topping the charts in the first practice session, before running fourth and fifth in the second.
Reigning champion Norris found it tougher going after gearbox issues ruined his first session before he was seventh quickest in the late afternoon.
Brown is insistent that Pistri and Norris can fight for the title again this year, provided McLaren give their drivers a car capable of winning races.
If that is the case, all eyes will be on how McLaren handle the pair.
The Woking-based outfit are one of the few teams that does not have a designated No.1 driver, with Norris and Piastri able to battle it out.
But commentary in Australia at the end of last year centred largely around a perceived favouritism to Norris, as the Englishman took the title from Piastri.
That was headlined by McLaren's request for Piastri to move over for Norris in Italy after a slow pit stop, along with claims from some quarters that the team disadvantaged the Australian.
"I thought a lot of it was very inaccurate," Brown said.
"The conspiracy theories and allegations that were made were so far wide off of the mark.
"You can't control people's opinions in sport, people will have their views.
"But it is shocking at times how uninformed people are in their allegations of what they think we are up to."
Piastri has made clear he intends to play the team game, noting not doing so would blow up his championship chances.
Brown also said there would be no significant shift from the team's approach this year, now famously coined as "Papaya Rules" after a team radio message last year, referring to the team's colours.
"At the end of the day we let both our drivers race hard, and we had two drivers that won seven races," Brown said.
"It came down to the last race of the year with (both drivers) a chance of winning the championship. I am quite proud of that.
"We were prepared for the consequence of someone not winning because of taking points off each other. As a racing team, we let both drivers race fairly and equitably.
"Of course we made mistakes along the way. That's racing."
Piastri was one of the few not to have a trip through the grass or gravel on Friday, with Max Verstappen suffering floor damage to his Red Bull late in the day.
The four-time world champion was third-quickest in the first session, before slipping to sixth in the afternoon.
New team Cadillac had a tough opening, with Sergio Perez suffering sensor issues and failing to log a flying lap in afternoon.
But the day was even worse for Aston Martin, with the team warning vibrations caused by power unit issues could stop them reaching the grid for Sunday's race.
Fernando Alonso failed to leave the pits for the first session and was 4.933 seconds off the pace for the second, while Lance Stroll managed just two laps in the late afternoon.
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