Dear @JJ061987,
Welcome back to the Wefugees Community and thank you for reaching out to us again.
I remember that we discussed one of your earlier questions here which was very similar, right? In accordance with my previous answer I believe that you can apply for a permanent settlement (Niederlassungserlaubnis) while you are in "Elternzeit" as you will return to your job after a few months and secure your livelihood with Elterngeld and Kindergeld in the meantime, am I correct?
The "Einbürgerungstest" (exam for the naturalization) contains almost the same questions as the so-called "Leben in Deutschland" exam (which is the most common certificate to prove the required "basic knowledge of the legal and social system and living conditions in Germany"). Therefore, the immigration office should accept this as well. However, I learned about some cases in which the civil servants at the immigration office were persistently asking for the "Leben in Deutschland" test. I recommend to give it a try nevertheless. Regarding your last question: No, you don't have to wait until your temporary residence permit expires before applying for the permanent one.
I hope this helps and please don't hesitate to get back to us with any further questions.
All the best,
Meike