Did barbarian invasions worsen or improve the condition of women in western Europe? What were gender roles in the Roman empire and early Germanic society?

The late Roman Empire was defined by a strict adherence to Christianity. In fact, the Olympic Games were briefly cancelled due to their connection with and ostensible worship of the pagan god Zeus.
Christianity without a doubt promoted the idea of specific and unequal gender roles in society. Since the disciples of Jesus were all male, only men were allowed to become priests. This idea reinforced separate gender roles, and men were granted more rights in the Roman Empire than women. Women were expected to handle the domestic tasks in the household. As the Roman Empire become older, the citizens became lazier and began outsourcing defense by hiring barbarian mercenaries to defend their border.
On the other hand, the barbarians and early Germanics did not embody these same gender roles. Women took leadership roles within barbarian society. So, in the short term, the barbarian invasions helped improve the condition of women in western Europe.
However, in the long run, the influence of Christianity prevailed, and there is evidence that the Dark Ages, which followed the fall of the Roman Empire, reinforced the Christian gender roles. So, the condition of women did not significantly improve following the barbarian invasions, primarily due to the mass adoption of Christianity.

An interesting question. The barbarian invasions of Rome represented a dramatic shift in European power dynamics. The Fall of Rome, marked by most as occurring in 476 CE when the last Roman Emperor was eclipsed by a German barbarian leader, led to important changes. Some of these changes involved gender roles.
Roman society was highly gendered. It was a true patriarchy, and the role of women was to be entirely subservient to men. Many are the Roman funerary monuments attest to the value of a woman's role being thoroughly domestic. Women were definitely second-class citizens, not even allowed to watch the gladiatorial combats close up, unless they were Vestal Virgins.
Barbarian societies and early Germanic society, in contrast, tended to be more egalitarian. Or at least, that's was how it was portrayed. There were cases of famous women leaders, such as Boudica, who led the Welsh in their resistance against the Romans in the 1st c. CE. But much of this may be exaggeration and mythology.
What is clear is Christianity had a major impact on the post-Roman Empire barbarian cultures, which only reinforced more traditional gender roles and the power of patriarchy. Still, one suspects there was more equality in these Dark Ages barbarian societies than in the Roman Empire.