As a parent and also someone with family members in hospice care, the problem is that the good quality care is prohibitively expensive for many, many people. People don't want to take time off to care for their loved ones just because, but because they can provide much, much better care at a lower cost.
Prohibitive medical and care costs are policy issues, that can be solved by many different methods, not necessarily requiring the same approach as parental leave.
Exactly. Parenting has significant impact on a child's development that cannot be replaced by nannying or daycare. The same can't be said for other types of care.