To elaborate on abx's comment: modding out by scalars, i.e., working in $PGL_3$ instead of $GL_3$, by definition the stabilizer of $p$ is the group of projective automorphisms of the curve $p=0$ which preserves the embedding of the curve.

If we assume that the curve is smooth and that there is a rational point, for example, if the field is algebraically closed, then this is an elliptic curve embedded by a degree 3 line bundle. For simplicity, assume $3 \ne 0$ in the field. Then the general theory identifies this subgroup with the 3-torsion of the elliptic curve, i.e., a finite group abstractly isomorphic to $Z/3 \times Z/3$.

If we do not assume there is a rational point (but still assume the curve $C$ is smooth), then we can say that $C$ is a torsor for an elliptic curve $E$ and identify the stabilizer with the 3-torsion $E[3]$ of $E$. This is a group scheme which is isomorphic to $Z/3 \times Z/3$ after passing to an algebraically closed field.

The extension of $E[3]$ by the scalars $G_m$ is an example of a finite Heisenberg group scheme, and one can use the Weil pairing to reconstruct it.

I'm not exactly sure what to say when $p$ is not smooth -- in the case of mild singularities probably one gets a similar description.