Japanese offices generally ask for a complete Japanese translation that names and is signed by the person who made it, rather than a translator with a particular accreditation. A translation signed by a named professional translator meets that format, but check the office's current guidance before you submit.
Japanese offices care less about letterheads than about two things: is the translation complete, and who made it? The Immigration Services Agency takes the same line for visa paperwork, asking for a Japanese translation of any foreign-language document with the translator identified. Our translations are signed by a named translator with their full name and contact details, so the office knows exactly who stands behind the wording.
Many parents translate the certificate themselves, and that's generally allowed. Where it goes wrong is detail: the registration district, the informant, each parent's occupation and any marginal notes from the registrar all need to appear. A short certificate leaves out the parents altogether, so order the full certificate before you start.