Maybe relevant: it seems it is possible for children born out of wedlock (to unmarried Taiwanese parents) to obtain a Taiwan passport, and possibly also citizenship (meaning HHR).
Your parents were married, but the authorities are rejecting your parents’ marriage certificate, so these other cases about children born out of wedlock might give you ideas about what alternative documentation you might use instead of your parents’ marriage certificate.
From that second thread, there is a comment by @tando that says:
The name on your cirth certificate is the same with his English name on his passport? His date of birth, address, etc. are recorded on your birth certificate? If so, I think it may be OK. If it is not so, or even it is so, Household registration office might request additional documents to prove your father daughter relationship, such as his affidavit, photos of your mother and father and you, old letters from/to him to/from your mother, DNA test, etc. You father should ask at Household Registration Office what they need to approve his appplication to acknowledge you.
Good luck, hang in there, and please let us know how it goes.