Thank you for giving me a birth?

"Giving a birth to me" is not just awkward, it is incorrect. One would say "Thank you for giving birth to me," without the indefinite article. One certainly could say that to one's mother, and possibly it could be said to both parents--but of course, not to the father alone. There are two holidays in the United States--Mother's Day and Father's Day--so I know of no locution which would express that idea. When addressing both parents, one might say "Thank you for bringing me into the world." Still, it seems odd--it's just not something I think you would hear a native-speaker of English say. There might be such a phrase in other English-speaking countries. I don't know if others observe a Mother's Day or Father's Day, and I've never heard of a Parent's Day.

The attitude toward parents and ancestors is not the same in the western world. One does not automatically assume that one owes something to one's parents for having given birth to one's self. After all, that might not have been such a favor. Ancestor worship just doesn't exist, although, of course, people of aristocratic classes might feel such gratitude--at the same time that they might say that great grandfather was a nasty old coot. Give the different attitudes toward parents and ancestors, it's hard to recommend an equivalent.