Sure, let’s expand on your proverb, and use it as a methaphor. The parents are responsible for the horse needing to drink in the first place. Even though they can’t make them drink, it is still their responsibility. When it doesn’t drink, that’s on them: they caused the situation leading to that failure
Resenting your parents for you being alive is one of the more juvenile hurdles that people tend to get over at some point in their teens. I am a little curious as to what your reasoning is for being unable to move beyond the perceived injustice of incarnation.
Your parents were born, just like you were. Is it your grandparents fault that they put that suffering on your parents who passed it on to you? What about their parents? And their parents? Which generation should have been sterilized to dodge your suffering?
Depending on your beliefs, your culprit is either Adam or Homo-Erectus. Better still, let’s blame the fish that walked, or the amoeba in the tide pool.
It’s time to get over it.