The rows threaten to take some of the sheen off a selection featuring a bumper crop of major stars including Brad Pitt, Johnny Depp, Kristen Stewart, Meryl Streep and Scarlett Johansson at the festival.
Having been pilloried for including only one woman director in 2018, the festival has two women out of 21 in the running for this year's Golden Lion top prize. Festival director Alberto Barbera said last year that he would rather quit the 11-day event -- where three of the last five Oscar best picture winners were premiered -- than give in to pressure for quotas.
This year the row over gender disparity has been compounded by a selection that has sparked particular fury -- that of Polanski, who was convicted for the statutory rape of a 13-year-old in 1978.
The inclusion in a sidebar event of US director Parker, who was embroiled in a rape trial while still at university, has also sparked criticism.
Melissa Silverstein, founder of Women and Hollywood, has accused the festival organizers of being "completely tone-deaf".
"We all know that the world has changed post #metoo," she told AFP. "Programming decisions are deliberate and very thought through. None of this information is hidden. The question I ask is, 'Is this a lack of awareness or deliberate indifference?'"
Barbera has defended the line-up, saying that "numerous films this year deal with the theme of the feminine condition in the world which, even when directed by men, reveal a new sensitivity."
He added this was proof that high-profile scandals had "left their mark".
The 76th Venice festival opens with Japanese director Hirokazu Kore-eda's "The Truth", starring French heavyweights Catherine Deneuve and Juliette Binoche, about a French cinema star whose decision to publish her memoirs prompts a mother-daughter reunion which turns fiery.
Saudi Arabia's Haifaa al-Mansour, the maker of the acclaimed "Wadjda", is one of the two women directors vying for the top prize. Her film "The Perfect Candidate" tells the story of a doctor trying to become her town's first female candidate in elections in the conservative kingdom.
Newcomer Shannon Murphy is the other female contender, with her Australian comedy-drama "Babyteeth".
Steven Soderbergh's take on the Panama Papers investigation, "The Laundromat", will also been premiered, while Pitt plays an astronaut in James Gray's highly anticipated sci-fi drama "Ad Astra."
Parker's 2016 debut film about a slave revolt, "The Birth of a Nation", was derailed after it emerged that he was accused of raping a fellow student, who later killed herself.
Parker was acquitted, but he later said "I look back on that time as a teenager and can say without hesitation that I should have used more wisdom."
Fellow black American director Spike Lee has vowed to travel to Venice to support Parker, whose film "American Skin" tells the story of a Marine veteran whose son is killed by police.
But it is the premiere of 85-year-old Polanski's historical thriller about the persecution of the French Jewish army officer Alfred Dreyfus, "An Officer and a Spy", which is likely to make most headlines.
With Polanski suing the Academy of Motion Pictures for stripping him of his membership, Screen Daily's chief critic Fionnuala Halligan was withering about his selection. She imagined festival director "Barbera, wandering the Lido hopelessly, singing the same mournful refrain... he can't find a female film director.
"So this year he's going to programme the new film by (a) convicted child rapist."
The festival opens as the accusations that sparked America's #MeToo movement are back in the spotlight. Harvey Weinstein, once one of the most powerful men in Hollywood, pleaded not guilty to two new charges of sexual predatory assault Monday, as a judge postponed his trial to 2020.
By Kelly Macnamara with Fiachra Gibbons