I think there is a little confusion about the birth registration requiring it to be registered.
I have looked at the requirements on the embassy in Helsinki and it does state the marriage be registered here at a Amphoe by stating a Kor Ror 22 is needed in the Thai version (translated by google).
"6. Copy of marriage certificate of parents In case of marriage abroad, please attach a document of marital status (Chor. 22) issued by the district according to your household registration."
Source: https://helsinki.thaiembassy.org/th/publicservice/66054-การแจ้งเกิดบุตร?page=5d80876d15e39c3354007b69&menu=5d80876d15e39c3354007b6a
On another embassy website it is not mentioned it on mentions a different statement from the mother.
""Mother's testimony record" (in case the father and mother did not register the marriage) [Download]"
Source: https://thaiembdc.org/th/thaibirthcertificate-thai-english/