Great question!
Sorry if it's offensive but I'm wondering how parenting works with Muslim father's. For example would a Muslim father change his daughters nappy or bath his children? It is that seen firmly as a woman's domain. And if he wouldn't then what would happen if the mother was hospitalised and there was noone else to take care of the children
Muslim fathers must parent their children, and yes that includes changing nappies and taking on their fare share of household chores and looking after children. The Prophet Muhammad did household chores and was always in the khidmah (which means service) of his family and wives. In fact he said: "The best of you are those who are the best to their wives, and I am the best of you to my wives.” (Narrated by al-Tirmidhi). My husband did all the nappy changes in the first few weeks post-birth before he went back to work. He was useless at night feeds though (although I was breastfeeding for the most part), but he is making up for it now, as the little one asks him to do everything (No, I want daddy to clean my bottom, NOT mummy hahaha
).
In some cultures, the rules are different, so I know some of my extended family members (regardless of faith) who are very cultural rather than religious, can't believe my husband changes my daughter's nappy for example (How can you be so cruel they tell me?
). I remind them of the Prophet and his actions, and if he can do chores, so can my husband.
Practically, I do end up doing more chores though (also not so much now as my husband's working pattern has changed) because I choose to work less hours per week as I want to spend them with my kids. My husband doesn't mind putting them in full time childcare, but I prefer not to. It is just a personal choice which means I have taken on more of the share of household chores. However, my husband has always paid for a cleaner, so I should cut him some slack? :)
Also, in Islam, it is the husband's responsibility to provide all the household income, including all the wife's expenses (the wife can work, but what she earns is for her own personal use :)). This does mean that many Muslim men do spend more time outside of their homes (away from their children) in comparison to their wives. But there is of course flexibility in this, so if the wife wants to be generous and share her income and contribute to the household income, she is free to do so. I don't personally - except maybe once a year when I am in a good mood 
.