In that scenario who deserves the title ‘parent’, the one who gave you life, or the one who stood by you through it?
I'm very familiar with cases like this, but if I'm to answer this question, I think it's going to go a bit beyond the definition of 'parent'.
Before I go beyond that, you should know that a parent is literally someone who gives birth to an offspring and cares for it. A parent can also be someone who's more of like a legal guardian
(like foster parents), and even someone that cares for you also fits this description.
But if I'm to go beyond the definition, then it's often makes more sense to say that the actual caregiver deserves the title far more, especially in a situation where it's becomes a problem. I have come across some many cases where biological parents comes out from nowhere, just to start fighting for a place in the life of a child that they never took care of. Of course, we can't alter the biological similarities
(like what they share), but everything that a child needs to grow is a bigger priority imo. And anyone who's able to give that, even if it's not their biological child, deserves to be called a parent.