Selection and Training Session

First note that all parabolas are similar, thus we may consider the parabola $y=x^2$. Let $a$, $b$, $c$, $d$, $e$ be the abscissae of the points $A$, $B$, $C$, $D$, $E$ respectively. We use the following easy lemmas.

Lemma 1. The chords $AB$ and $CD$ of the parabola are parallel if and only if $a+b=c+d$.

Lemma 2. The points $A$, $B$, $C$, $D$ of the parabola (at least three of them let be different) are concyclic if and only if $a+b+c+d=0$.

From Lemma 2 it follows that in addition to $A$, $B$, $E$ the circle $S_1$ has one more common point $F_1$ with the parabola, the abscissa of $F_1$ being $f_1=-a-b-e$. Similarly, $S_2$ has common point $F_2$ with the parabola, its abscissa $f_2=-c-d-e$. Since $AB\parallel CD$, we have $a+b=c+d$, so $f_1=f_2$. This means that $F_1=F_2=F$ belongs to the parabola.