On a continuous S\'ark\"ozy type problem

We prove that there exists a constant $\varepsilon > 0$ with the following property: if $K \subset \mathbb{R}^{2}$ is a compact set which contains no pair of the form $\{x, x + (z, z^{2})\}$ for $z \neq 0$, then $\mathrm{dim}_\mathrm{H} K \leq 2 - \varepsilon$.