It's the last bit of "power" either of them have left, and those with little to no power will cling hard and fast to any little crumb they have. Power over each other because the "victor" the one who doesn't go first will for ever and ever have the chilling tale of woe to recite over and over to all who will listen about how their ex, "signed over their own child." And power over you, I'm sure because hey, in some way they think, "still MY kid," if you get what I mean.