The petition, which has been signed by 100,000 people, was handed over on Thursday - International Women’s Day.
It was organised in response to the #MeToo campaign and revelations of sexual assault and harassment in Parliament itself.
Several MEP assistants have alleged they have been victims of such abuse, with some also complaining that their claims have not been taken seriously by the authorities.
The petition presented to Tajani demands changes in the culture and rules of Parliament to prevent future harassment and calls on the European Commission to propose a directive on violence against women.
Reintke, the Greens/EFA group women’s rights and gender equality spokesperson, also revealed how she had fallen victim to sexual harassment.
She said the attack happened recently while she was walking home after a night out with friends.
The MEP said a man approached her and put his hand between her legs. “I screamed and tried to run after him. The one thought in my head was to scare him enough so that he would not do this to any woman again. It was outrageous and I was in complete shock.”
The police quickly arrived in three cars and asked if she needed psychological help.
She says, though, that too many women are still “ridiculed” and “not taken seriously” when they make allegations of sexual harassment.
Reintke, a German MEP, added, “Me Too isn’t just media hype. Together, we are challenging abuse of power and demanding real change.
“In the wake of the revelations of sexual harassment within the European Parliament, there was cross-party consensus that change was needed. But warm words are not enough. When women are unsafe in their own workplace there has to be concrete action.
“More than 100,000 people have now backed our calls for zero tolerance towards sexual harassment. The President of the European Parliament must show that he is listening and ready to act.”
The petition was created and promoted with Period, a Brussels-based feminist network.
To mark International Women’s Day, the Commission released a report on gender equality that it says highlights where action is needed “to do more to ensure that we achieve real change when it comes to ensuring equality between men and women.”
The report shows that while European women are better educated than men, they remain largely under-represented in decision-making positions in companies and still earn 16 per cent less than men on average across the EU.
The Juncker Commission said it has made “tangible efforts” to address gender equality, including a proposal to improve the work-life balance of working families, an action plan to tackle the gender pay gap and funding and awareness-raising actions. However, a Commission spokesperson conceded, “But we need to do more.”
On Thursday, Commission first Vice-President Frans Timmermans said, “The issue of gender equality is high on the agenda, but progress is still slow on the ground. To achieve real change for women we need to turn awareness and intentions into action.
“That means adopting the new legislation the Commission has proposed on work-life balance, joining the Istanbul convention and implementing the policies we have already agreed on to tackle the gender pay gap and fight violence against women.”
European gender equality Commissioner Vera Jourová added, “Gender equality is not just about women. It is about our society, our economy and our demography. We want to guarantee that women are truly equal to men in front of the law.
“We will also continue work to empower women, so they can make their own choices when it comes to their careers and their families.”
Another study published on Thursday said that more women in digital jobs could create an annual €16bn GDP boost in the EU. However, only 24.9 per cent of women in higher education graduate in tech-related fields.
To increase the participation of women in the digital sector, the Commission outlined an initiative to increase women’s participation in the digital sector.
Elsewhere, European Economic Social Committee (EESC) President Georges Dassison said this year’s Women’s Day “finds not just Europe, but the whole international community, at a critical economic and political juncture with violations of women's rights on the rise.
“In various parts of the world where there are conflicts, migration and displacement, and wherever the effects of poverty and climate change are at their most severe, women and children are among the groups most vulnerable to discrimination and violence.
“Europe remains a pioneer in issues of gender equality, something we should be proud of. Sixty years ago, gender equality was enshrined in the Treaty of Rome as one of the EU’s fundamental values.
“However, the significant progress made is not consistent. For instance, despite the women’s employment being at a historic high, unemployment among women remains significant, especially in southern Europe.
“In all the EU countries women earn on average 40 per cent less than men, while the disparity in pensions is unchanged at 38 per cent. At this rate, it will unfortunately take another century for income disparities between men and women to be eliminated. Women also continue to face a glass ceiling blocking their access to managerial and leadership positions, and they are still under-represented in politics.”
Dassison added, “Much still needs to be done within and outside of Europe. The EESC will persist with its efforts to protect women's rights, also in the workplace, to eradicate all forms of inequality, to combat misogyny and all forms of violence against women, and to widen equal opportunities and improve family benefits, demonstrating that women are equal to men in every sphere.”