  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 $\mathcal{T}(\mathfrak{p}^r)$.
