What I said is not incorrect, and you can read that for yourself at either the Stonewall website under parental responsibility and in the lesbian parents' guide at the Rights for Women website (the name seems ironic given the comments on this thread).
The argument as to why gay marriage promotes heteronormativity has been stated many times by numerous lesbian feminists. Society (and your argument is a good example of this) wants to promote the heteronormative, heterosexual two parent family as a means to various social ends, gives two parent families extra rights, and uses the existence of gay marriage as a moral justification for why this is okay. I am not arguing against gay marriage, but I certainly resent it being used as a justification for why heteronormative families (mostly families with a man in them) are entitled to more from society than other families (mostly families with only women in them).
Your argument is that people with caring responsibilities will be treated less negatively if men are half the carers. To achieve this outcome, it would require almost all mothers to enter into a relationship with a man and then privately negotiate so that he does half the caring so that she, and women more generally, are not seen as a risk. So to get equality you require women to agree to heteronormativity.
We live in a society where most women choose whether or not they want to be pregnant. As pregnancy carries health risks and is, at the very least, a bit of a hassle, the vast majority of women get pregnant because they intend to spend some time being with their child and be in that child's life. Who else that woman is intending to relate to in the arrangement of the care of that child varies. So a family is almost always mother plus children, but does not neccesarily have any man involved at all. Your choices to change that are a. force women to get pregnant and hand over the baby to a man or b. force women into relationships they don't want. Women not having rights in this country led to a little forced pregnancy and a lot of forced relationships.
I don't know who does secondary care in countries like Sweden who give a range of secondary carers leave, but the global pattern is that it is women who do a lot of secondary care. To not give those women leave is to say that to achieve the aim of women being treated equally, you are going to treat those women negatively now by making them fit in care around inflexible working patterns, in the hope that this will lead to a positive situation for women in the future.
We are never going to get into a situation where men as a group do as much childcare as women as a group because many women do not want to get rights or equal treatment by privately negotiating their finances and childcare arrangements with a man in a long term heteronormative relationship, but they still want kids. Their negotiation is with society - in providing affordable, high quality child care, working rights for both people with caring responsibilities and for the secondary carers (often women) they have chosen to raise children with.
Nobody else in the UK except mothers is expected to gain their rights by entering into a sexual relationship with someone from a different political class, move in with them and then come to a private agreement about what the other person is or is not prepared to do.
All women are discriminated against, but mothers in heteronormative relationships have more rights and are less discriminated against outside the home than other mothers. The price of that is the risks they face within the home - a situation many women do not want to enter or are desperate to leave. The way to improve things for all mothers is to provide decent childcare, strengthen carers' rights and maternity rights in the workplace, and give rights (including workplace rights) to whoever it is who supports children and mothers, regardless of whether or not that person is the other parent.
And if we want a society where men do equal caring to women, and it isn't actually about men controlling women's reproductive labour or women in LTRs with the father wanting to feel morally superior to other mothers, there are plenty of old people men could be providing unpaid care for.