you are wrong...if your name on a BC at a time a baby born (one of the parents maybe oversea at that time)...you're still the parent and yes...it means something. The only prove is through a DNA test if needed.
it's easier to go through the US Embassy than the Thai Authority for the parental rights. Now for the parts you want to stay longer for whatever reason, you have to follow the thai rules. No way to avoid it. It takes time if you must do.
but I got it, go to the US Embassy for the assistance if needed. They would help you get DNA test if that's the case. My baby has my blood, Thai authority can't deny my child rights of the father. That's the international laws.