Teachers at a private school do not require a State of California Teaching Credential. From the FAQs:
Operators of private schools are required by law to file a Private School Affidavit. In addition, the law provides an exemption from compulsory public school attendance for children who are being instructed in a private, full-time day school (EC 48200, 48220, 48222), so long as certain conditions are met: instruction must be provided by persons capable of teaching; and instruction must be in English and offered in the several branches of study required to be taught in California's public schools. In addition, attendance must be kept in a register that indicates every absence of the pupil from a half day or more of attendance. For the exemption from the compulsory attendance law to be valid, the attendance supervisor of the public school district must verify that the private school has filed the annual Private School Affidavit required by EC Section 33190. If the school has nonprofit tax status, the eligibility of that school's students, teachers and other educational personnel to participate in various federally funded programs is contingent upon the annual filing of the Affidavit.
The “absence of half day or more” of attendance rule is also stupid: I’m not a homeschooler but I fully support them (and I know several), and I know that with the one-on-one tutoring they do that they are often done with a whole day’s schoolwork in less than half a day.