The short answer
Without legitimation, an unmarried father in Thailand generally holds no parental power — no right to decide on schooling, medical care, travel or residence, and no standing to apply for custody.
Three routes to legal fatherhood
Marriage to the mother; registration of legitimation at the district office with her consent; or a court order. The third is available where consent is refused or the mother cannot be located.
What legitimation does not do
It does not by itself transfer custody. It makes the father a legal parent, which is the precondition for applying.
Obligations follow rights
Legal fatherhood also brings a support obligation. That is worth understanding before starting, not after.