The differential equation of all parabolas whose axes are parallel to the axis of y, is

$\frac{d^3 y}{d x^3}=0$

The equation of the parabola having their axes parallel to y-axis is given by

$y=a x^2+b x+c$

$\Rightarrow \frac{d y}{d x}=2 a x+b \Rightarrow \frac{d^2 y}{d x^2}=2 a \Rightarrow \frac{d^3 y}{d x^3}=0$

This is the required differential equation.