parabola passes through $4$ points
You rightly noticed that the four points are symmetrical around $y=x$.
So change the variables accordingly
$$
\left\{ \matrix{
\xi = y - x \hfill \cr
\eta = y + x \hfill \cr} \right.\quad \Leftrightarrow \quad \left\{ \matrix{
x = {{\eta - \xi } \over 2} \hfill \cr
y = {{\eta + \xi } \over 2} \hfill \cr} \right.
$$
The four points become $$ \left( {1,3} \right),\left( { - 1,3} \right),\left( {1,7} \right),\left( { - 1,7} \right) $$
The parabola's axis is the $\eta$ axis, so its formula is $$ \eta - a = k\xi ^{\,2} $$
In any case it is clear that, since the points form a rectangle, there cannot be a single parabola passing through all of them.