There is a solution using the concept of envelope. Indeed, parabola $P$ with equation $y=x^2$ can be considered as the envelope of all possible ping-pong balls with generic equation:
where we have to check that $f(r)=r^2+\frac14$.
The classical method for the determination of envelopes is obtained by working with a system of 2 equations, the initial one (1) and the equation obtained by differentiating it with respect to the parameter, here $r$, which is :
and plugging this expression into (1), we get
Replacing now $y$ in the LHS of (3) by $x^2$ in (4), we get:
which is equivalent to:
We will not solve this differential equation.
It is sufficient to check that $f(r):=r^2+\tfrac14$ is a solution.