jmc

Adding the equations, we get $$x+y=x^2+15x+32+y^2+49y+593,$$or $x^2+14x+y^2+48y+625.$ Completing the square in $x$ and $y,$ we get $$(x+7{)}^2+(y+24{)}^2=0.$$We can check that $\boxed{(-7,-24)}$ lies on both parabolas, so this is the point of tangency.