I used to oversee our leave department at work in California, so I know just how crazy confusing leaves in California can be.
First, you need to mentally separate out PAID leave vs. PROTECTED leave.
As a new parent, you are entitled to 12 weeks of PROTECTED leave in the 1st year after your baby's birth. This leave is protected, meaning your employer can't fire you for taking it. It has absolutely no pay attached to it. It is a federal law called the FMLA. It only applies if you've worked at your current employer for at least 12 months.
[Note, as a pregnant woman, it gets WAY more complicated. I'm presuming you are the non-birth-giving parent. If you gave birth, I can explain how various leave laws work to give you about 45 weeks of protected leave in more detail but it will give you a headache.]
Second, as a Californian, you are entitled to 6 weeks of PAID family leave (PFL). This leave is paid by the State of California, generally about 55% of your normal take-home pay, and has NO PROTECTION. That means if you choose to use it, the State will pay you, but your employer can fire you. There is a 2 week waiting period to get the pay, so you actually have to take 8 weeks off to get paid for 6. However, you can't get paid more than 100% of your salary, so if your employer paid you full pay while you took time off, you can't take the pay from the State. But, if you take the California pay during the 12 weeks of protected FLMA leave, then you have both protected leave and paid family time.
Really easy to understand, right? :)
So, yes, as a new parent (non-birth giving), you get 12 weeks of leave. California will pay you for 6 of those. If your employer paid you for 4 weeks of leave, and then you went back to work, you can take 8 more weeks of leave prior to the baby's 1st birthday, and the State will pay you, and your employer can't fire you. The 4 weeks you took paid can count towards your 2 week waiting period, too.