If father gave up parental rights would he still have to pay child support in CA state

That depends. The parents could come to whatever agreement they choose to with regard to dad's "rights", but technically speaking, child support is not waivable. You can set the support payment at zero, which eliminates the problem short -term, but leaves Dad on the hook later if the situation arises that Mom asks for it. Seek legal counsel in your area for more details. Good luck to you!