It is not absolutely irreducible: over the algebraic closure, it is the direct sum of $V^{Fr ^i}$ ($0\leq i \leq e-1$), where $\operatorname{Fr}\colon{\mathbb F}_q\rightarrow{\mathbb F}_q$ is the map $x\mapsto x^p$.
I think $e$ being odd or even does not matter: $V$ is irreducible over ${\mathbb F}_p$. The restriction of scalars may be viewed as follows. First, $V$ viewed as a ${\mathbb F}_p$ vector space has dimension $3e$, and one has an embedding ${\mathbb F}_q\subset\operatorname{End}_{{\mathbb F}_p}(V)$. The algebra $M_3({\mathbb F}_q)$ is simply the commutant of this copy of ${\mathbb F}_q$ in $\operatorname{End}_{{\mathbb F}_p}(V)$ and is spanned as a ${\mathbb F}_p$ vector space (a small check) by elements of $\operatorname{Ad}(\operatorname{SL}_2({\mathbb F}_q))$.
Suppose $W\subset V$ is irreducible for the adjoint action of $G=\operatorname{SL}_2({\mathbb F}_q)$. Then $W$ is stabilised by $M_3({\mathbb F}_q)$; in particular, $W$ is an ${\mathbb F}_q$ subspace of $V$ (now viewed as a ${\mathbb F}_q$ vector space). But $\operatorname{SL}_2({\mathbb F}_q)$ acts irreducibly on $V={\mathbb F}_q ^3$, hence $W=V$. [This irreducibility fails of course when $p=2$].
[Edit] I think I can find a proof that the ${\mathbb F}_p$ algebra $R$ generated by $\operatorname{Ad}(\operatorname{SL}_2){\mathbb F}_q)$ is all of $M_3({\mathbb F}_q)$. First of all, $V \otimes _{{\mathbb F}_p}{\overline {\mathbb F}_p}$ is the direct sum of $W_i=V^{Fr ^i}$ as $i$ varies from $0$ to $e-1$, and is semi-simple. Therefore, $V$ is semi-simple as an $R$-module. Hence $R$ is semi-simple (the action of $R$ being faithful on $V$). Therefore, $R$ is a product of simple rings $R_i$, each $R_i$ of the form $M_{d_i}({\mathbb F}_{q_i})$ a matrix algebra over a finite field containing ${\mathbb F}_p$.
The module $V_i={\mathbb F}_{q_i}^{d_i}$ is absolutely irreducible for $R_i$. However, $V$ over the algebraic closure, is a sum of irreducible three dimensional representations $W_i$. Therefore, each $V_j$ is some $Wi$ over the algrbraic closure, and is hence three dimensional: $d_i=3$. Moreover, since $R$ commutes with the action of $F={\mathbb F}_{q}$ on $V$, each ${\mathbb F}_{q_i}$ contains $F$. We have thus,
$$M_3({\mathbb F}_q)\supset R\supset R_i=M_3({\mathbb F}_{q_i})$$ which shows that $R=M_3({\mathbb F}_q)$ as needed.
