I got a w2 for my paid family leave for CA from a 3rd party. Do I paid CA state taxes on this?

No, for California state taxes, PFL benefit payments are not reportable pursuant to Revenue and Taxation Code Section 17083.  In TurboTax, you will be able to indicate that it was Paid Family Leave after you enter the W2 information.  (If you did not see it, go back through the section again.)

When i enter the w2 information and on the 3rd page I fill out it was for PFL, the amount goes down in my State for what I owe, but then when I go to the next page to continue, my State Tax owed goes back up

Remember that the refund meters are live calculations and will go up and down as you make your entries.  When you get to the California state section, you will have to verify the Paid Family Leave.

See below.