the equation for vertical parabolas makes sense as the tangent at the vertex of vertical parabolas has a slope of zero, so derivative of $y=ax^2+bx+c$ which is $2ax+b$ should be 0 when we want to find out the x co-ordinate of the vertex.
but since the slope of y-axis is undefined, so derivative of $x=ay^2+by+c$ which is $2ay+b$ should be equal to undefined if we have to find out the y co-ordinate which seems weird
and further we use $y=-b/2a$ while actually finding out the y co-ordinate for the vertex. why and how is $y=-b/2a$ in this case?
If $x = ay^2 + by + c$ (assuming $a\neq 0$ so that this really is a nondegenerate parabola), then you can find an expression for the slope using implicit differentiation:
$$x = ay^2 + by + c$$ $$\frac{d}{dx}\Bigg(x\Bigg) = \frac{d}{dx}\Bigg(ay^2 + by + c\Bigg)$$ $$1 = 2ay\frac{dy}{dx} + b\frac{dy}{dx} + 0$$ $$1 = (2ay + b)\frac{dy}{dx}$$ $$\frac{dy}{dx} = \frac{1}{2ay + b}$$ The vertex would be where the tangent is vertical, so that this expression for slope is undefined, i.e., where $$2ay + b = 0$$ $$y = -\frac{b}{2a}$$ which is well-defined since we assume $a\neq 0$.