Let $\mathfrak{n} = \mathfrak{p}^r$ be a prime power ideal of $\mathbb{F}_q[T]$ with $r \geq 2$. We study the rational torsion subgroup $\mathcal{T}(\mathfrak{p}^r)$ of the Drinfeld modular Jacobian $J_0(\mathfrak{p}^r)$. We prove that the prime-to-$q(q-1)$ part of $\mathcal{T}(\mathfrak{p}^r)$ is equal to that of the rational cuspidal divisor class group $\mathcal{C}(\mathfrak{p}^r)$ of the Drinfeld modular curve $X_0(\mathfrak{p}^r)$. As we completely computed the structure of $\mathcal{C}(\mathfrak{p}^r)$, it also determines the structure of the prime-to-$q(q-1)$ part of $\mathcal{T}(\mathfrak{p}^r)$.

21 pages, to appear on Research in the Mathematical Sciences