I will give it a try, based on Day's idea. Let $X$ be a two-dimensional Banach space with the given property and denote by $B_X$ its unit ball. Consider the ellipsoid of maximal volume (denoted by $B_2$) contained in $B_X$ (the John's ellipsoid) and denote by $\|\cdot\|_2$ the induced Euclidean norm. The goal is to show that $B_X=B_2$.

From John's theorem concerning the ellipsoid of maximal volume (or Loewner's Lemma for two dimensions in Day's paper), it follows that $B_X$ and $B_2$ have at least four contact points. Unless $B_X=B_2$, the contact points cannot form a dense subset of $B_X$. Assuming $B_X\neq B_2$, we can find contact points $x$ and $y$ such that $\displaystyle\frac{x+y}{\|x+y\|}\in B_X$ is not a contact point. Hence $\|x+y\|<\|x+y\|_2$. Therefore:

$$
4=\|x+y\|^2+\|x-y\|^2<\|x+y\|_2^2+\|x-y\|_2^2=4
$$

This is a contradiction, hence $B_X=B_2$.

Parallelogram law for vectors of equal length. $\endgroup$