Of course a non-custodial parent should have to pay child support for a child if they want to be a part of the child's life. Being a parent is not about getting what you want, it is about doing what is best for the child. If you want to be part of that child's life, you are responsible for its well-being and upkeep.
When in a relationship, each parent makes a contribution to the upbringing of that child in both a financial and moral role, why should that change because the parents are no longer living together?
The parent should pay a reasonable amount for the amount of time that they are not financially supporting their child.
Financial support and parental support are not mutually exclusive events
Children should not be punished because someone decides that they don't want to "pay" to support that child. Children have expenses and it takes two to create that child. Both parents should take responsibility to help raise that child. That said, it's wonderful when both want to be active in a child's life, at the same time, children have needs that cost money, and if you tangoed, then you need to help to raise that child and provide for all their needs, including medical, housing, transportation, food, clothing, etc.
The non-custodial parent needs to pay child support in all circumstances. Even if the custodial parent doesn't want it, the money can be put towards a fund where the child can access when he/she turns 18. You brought the child to the world, it is your responsibility to provide and care for the child. Child support is the LEAST amount you can do.
When it comes to this there are often no equal rights considered and the child is always handed off to the mother for the father to pay, even if they were never married and the condom broke and the whole time the women get to chose stuff like abortion and the father never has a say beyond the point past conception. If women want equal rights then men should get too.