The idea of thinking of groups women against men is collectivist in my view and irrelevant. Your suggestions creates more equality between men and women on a statistical basis, but create huge problems for parents compared to people without children. To tell a fresh mother, not only she has damaged her career, but the career of her husband, too, is not exactly something she will appreciate, even if the profiteer in her husbands company is a woman.
Situations are different from family to family, but often fresh fathers work more than before, to get extra pay for extra hours. This possibility would be taken away completely.
As well currently many women expect, that family plans are coordinated with their career plannings, as there are times, when a leave fits better, and times, when it fits worse. With a forced paternity leave, every couple would have to consider both careers carefully. Already now university educated people have less children in Germany, and probably in most European countries, than none-academics. This last aspect of a inhomogenous career is especially a problem for high qualified people, who have something one can reasonably call a career. Academic women are more over disproportionally in jobs like teaching and other public jobs, where a leave has less negative effect than academic men, so that this fit-to-career planning would have to be increased.

As for providing more money to parents, sure, though I would say Germany needs to spend equal attention to providing some things (e.g. day care) for free.
I'm fundamentally opposed to free day care. This is an assault on the classical family. That is not freedom to chose between staying at home and letting care for the children, when one decision is subsidiesed with hundreds of Euros per month and the other not. Parents should be free in such a decision, but I personally would of course favour the model of the classical family over unsymbiotic forms of family live.
And of course plenty of proprietary and organic solutions may be there, which can only be applied from case to case like caring grandparents, neighbours, telework, workplace built-in-childcare, or even a nanny ,into which is not tapped by any one-size-fits-it-all assumption.

Given that in Germany well-off families with children are strongly overtaxed relative to none-children tax payers, and that the budgets are negotiated with a certain focus on previous budgets, so that childcare is in confilct with any direct monetary payments to parents, it means parents with differnt solutions are the ones who practically pay for those, who send their children to child-care or for extended parential leave (especially the paid one, as classical families will not get a cent for their second or third or fourth child for the paid maternity leave).

when change is needed to deal with new challenges (such as a demographic downturn)
If this measures are ultimatively not financed by taking it from elsewhere than from families, these suggestions will make the demographic challenges more pronounced than the current challenges. And the demographic challenges are different in various countries. E.g. I don't see any need to stimulate fertility in most west European countries, while the Balitic states face a population implosion. One answer to these very different countries is not the right answer, regardless how this answer looks like.

The only point of this freedom
Freedom is a value by itself, and therefore the default. You need a substancial cause to diminish it.

, for the men's side, is to retain a comparative advantage in the labour market over women (or retain parity with singles).
As I wrote, there are other considerations than pure men-women issues. It is a point, but not the only.

Der Amerikaner ist die Orchidee unter den Menschen
Volker Pispers