Let $P(x_1,y_1)$and $Q(x_2,y_2),y_1<0,y_2<0$be the ends of the latus rectum of the ellipse $x^2+4y^2=4$.The equations of parabolas with latus rectum $PQ$are

$\frac{x^2}{4}+\frac{y^2}{1}=1,a=2,b=1$

The ends of latus rectum $PQ$are

$(\pm \sqrt{a^2-b^2},-\frac{b^2}{a})$

$P=(-\sqrt{3},-\frac{1}{2}),Q=(\sqrt{3},-\frac{1}{2})$

The midpoint of $PQ$ is $M(0,-\frac{1}{2})$

which is the focus of the parabola.

The vertex of the parabola is $V(0,-\frac{1}{2}\pm \frac{\sqrt{3}}{2})$

$\therefore$ The parabolas are $x^2=\mp 2\sqrt{3}(y-(-\frac{1}{2}\pm \frac{\sqrt{3}}{2}))$

i.e., $x^2-2\sqrt{3}y=3+\sqrt{3}\text{ and }{x}^2+2\sqrt{3}y=3-\sqrt{3}$