The UN has said media restrictions and violence meant the environment was not conducive to free, credible elections.
Unrest started in April after President Pierre Nkurunziza said he would run for a third term - something protesters say is illegal.
The president says he is entitled to a third term because he was appointed for his first term, not elected.
The presidential election is scheduled for 15 July. East African leaders have called for a further two-week delay.
Africa news highlights: 7 July
The electoral commission spokesman told the BBC turnout for the parliamentary poll had been low in the districts of Bujumbura where there had been protests, but that in some provinces outside the capital it was as high as 98%.
The ruling party - the CNDD FDD - was ahead in every province of the country, Burundi's electoral commission announced.
They won 77 out of 100 elected seats in parliament, AFP news agency says.
The BBC's Maud Jullien says all of the country's private broadcasters have closed in recent months, and many civil society leaders have left the country claiming their lives were under threat.
At least 70 died and 150,000 people have fled the country.
Reporting on a coup and a crisis... using a music-sharing site
10.4m population
50 years - life expectancy for a man
2nd poorest country in the world
85% are Hutu, 14% Tutsi
300,000 died in civil war

Up to a dozen A-29 Super Tucano aircraft would be sold to Nigeria to help fight Islamist militant group Boko Haram, unnamed US officials said.
The deal, which is not yet official, will require approval from Congress.
Boko Haram's deadly insurgency has displaced more than two million people.
The deal, said to be worth up to $600m (Â£490m), was agreed by the Obama administration, but was reportedly halted on the day it was due to be sent to Congress, after a catastrophic incident involving the Nigerian military.
About 90 people, mainly women and children, were killed in January when the Nigerian Air Force mistakenly bombed a camp in the country's north-east, which was hosting thousands of those who had fled Boko Haram.
An aid distribution was taking place at the time of the attack, according to medical charity MSF.
The Nigerian government indicated last month that the deal might be back on, following the first phone call between President Muhammadu Buhari and President Donald Trump.
"President Trump assured the Nigerian president of US readiness to cut a new deal in helping Nigeria in terms of military weapons to combat terrorism," Mr Buhari's office said in a statement.
The US congressional source said human rights concerns remain, despite support for the sale from some lawmakers, Reuters news agency reports.
The US Air Force described the A-29 aircraft as a "game-changer" when they were deployed in Afghanistan in 2016.
They can be armed with two wing-mounted machine guns and can carry up to 1,550 kg of weapons.
But the aircraft that would be sold to Nigeria come with a "very basic armed configuration," one of the unnamed US officials told Reuters.

Zoe Gregory, 26, is accused of sending a message claiming an explosive had been left at Ormiston Victory Academy in Costessey, Norfolk, on 9 February.
The 16-year-old girl's home was raided and she was questioned over the threat by police.
Ms Gregory has been bailed to appear before Norwich magistrates on 14 April.
She has been charged with communicating false information and unauthorised computer access.
Ms Gregory, of Blackhill Wood Lane, Costessey, has been dismissed from her role, said a school spokeswoman.
The ruling party in Burundi has won the parliamentary election boycotted by the main opposition parties.
[NEXT_CONCEPT]
The Trump administration plans to sell military planes to Nigeria despite concerns over rights abuses and a botched air strike that killed scores of civilians in January, US media say.
[NEXT_CONCEPT]
A teaching assistant has been charged with using a pupil's email account to make a bomb threat against the school where she worked.