We use data from the 2019 EU Labor Force Survey to study gender and parenthood gaps in two dimensions of flexibility in working time arrangements in 25 European countries. We find that overall in Europe, there is no statistically significant gender difference in access to flexible work arrangements. However, women are less likely than men to have flexible working hours in the Central-Eastern and Southern European countries, whereas this gender gap is reversed in Continental Europe. At the same time, women are less likely than men to face demands from their employers that they work flexible hours. We also find that both mothers and fathers are more likely than their childless colleagues to have access to flexible working hours, but that fathers' workplaces are more likely than mothers' workplaces to demand temporal flexibility from employees. In addition, we find that working in a female-dominated occupation decreases the probability of having access to flexible work arrangements, and that this effect is stronger for women than for men. At the same time, we observe that both men and women who work in female-dominated occupations are less exposed to flexibility demands from employers than their counterparts who work in male-dominated or gender-neutral occupations. Finally, we find that compared to employers in other Europeans countries, employers in the Central and Eastern European countries are less likely to offer flexible working hours, especially to women, and with no additional flexibility being offered to parents; whereas employers in Continental and Nordic countries are more likely to offer flexible work arrangements, and with no gender gap.