Arithmetic circuits over $GF(p)$ for computing $x \bmod 2$ given input $x \in GF(p)$
It's not clear if this is directly useful, but lower bounds are known for depth-3 circuits computing an $n$-variate version of your function over $\mathrm{GF}(p)$ for $p \neq 2$. See Theorem 3.7 in the survey of Shpilka and Yehudayoff for details and references.

I don't believe lower bounds of the form $\omega(\log d)$ are known for general circuits computing an explicit univariate polynomial of degree $d$, so proving an $\omega(\log p)$ lower bound for your problem will likely be difficult.