Under Parente's leadership, Petrobras posted its best quarterly results in five years, and the company's stock price doubled.
Brazilian oil workers began a 72-hour strike on Wednesday in a new blow to President Michel Temer following a nationwide trucker protest that has strangled Latin America's largest economy for over a week. Many analysts credit Parente with turning the company around since he took over about two years ago.
In a statement, Petrobras said the company's board of directors was meeting on Friday to study the possibility of naming a temporary replacement.
Petrobras shares were down 13 percent in early afternoon trading, pulling Brazil's wider Bovespa index into negative territory.
"We will maintain the economic policy that took the company out of the red in the last two years and made it once again one of the most respected in Brazil and overseas", he said.
In the aftermath of the strike, Temer hinted that he might restore government price setting, but he later backtracked.
In his resignation letter, Parente said the strike had set off an intense debate over Petrobras' pricing policies but little reflection about the realities of world fuel prices.
Unions representing oil workers said they were demanding the resignation of Chief Executive Pedro Parente. Mr Temer is seeking a new Petrobras CEO who will be more focused on the nation's development and less on the market, said Beto Mansur, a lawmaker close to the president, in an interview. Also, the government convinced Petrobras to change its pricing policy, so starting June, the company will only adjust prices monthly instead of daily.
The company risks becoming a political football in October's presidential election, with far-right candidate Jair Bolsonaro's key economic adviser calling for its privatisation, while leftist Ciro Gomes has warned investors to avoid state energy sector assets because he would expropriate them.
Parente said the company will not subsidize the price of diesel and will not incur losses, since it will be reimbursed by the government, "in a modality still to be defined".
Petrobras announces additional cut in diesel to 2.0316 reais a liter under subsidy program with govParente has always been at the center of finance in Brazil.
Workers pedal their bicycles past a banner with a message that reads in Portuguese: "Get out Parente", in reference to Pedro Parente, the president of the state-owned oil company, at the entrance of a Petrobras refinery during a strike in Duque da Caxias, Brazil, Wednesday, May 30, 2018.
It was that fuel policy, which matched local fuel prices to global rates, that came under fire during a massive truckers strike that wreaked havoc on Latin America's largest economy.
"The composition of the rest of the members of the executive board remains the same", the company said. Laborers were back on the job at 95 percent of the company's units, the company added.