Let $d(c)$ denote the Hausdorff dimension of the Julia set $J(z^2+c)$. First, we will investigate the derivative $d'(c)$, for real $c$ converging to a parabolic parameter $c_0$. We will prove that $d'(c)$ tends to infinity, when $c\nearrow1/4$. Moreover, we will see that $d'(c)$ tends to a constant or minus infinity depending on the value $d(c_0)$, where $c_0$ is a parabolic parameter with two petals. Next, we will investigate the directional derivative at $c=1/4$, in all directions, except the direction related to the parabolic implosion phenomenon.