Let $\varphi$ be an automorphism of the multiplicative group $\mathbb{F}_p^*$. Then 
$$x\sigma_{a,p}\varphi = (a^x)\varphi = (a\varphi)^x = x\sigma_{a\varphi, p}.$$
So the permutations $\sigma_{a,p}$ form a coset of the group of automorphisms of the cyclic group $\mathbb{F}_p^* \cong C_{p-1}$.  

Now when $p-1=m=2u$ with $u$ odd, then $C_m = C_2 \times C_u$ and it is easy to see that every automorphism of $C_m$ is an even permutation. When $m$ is divisible by $4$, then the automorphism sending $c$ to $c^{-1}$ is odd, since it is a product of $(m-2)/2$ transpositions, and $(m-2)/2 $ is odd.  

What I don't see just now is when one (and then all) $\sigma_{a,p}$ is odd or even in the case $p\equiv 3 \mod 4$.