  We consider the parametric estimation of the volatility and jump activity in
a stable Cox-Ingersoll-Ross ($\alpha$-stable CIR) model driven by a standard
Brownian Motion and a non-symmetric stable L\'evy process with jump activity
$\alpha \in (1,2)$. The main difficulties to obtain rate efficiency in
estimating these quantities arise from the superposition of the diffusion
component with jumps of infinite variation. Extending the approach proposed in
Mies (2020), we address the joint estimation of the volatility, scaling and
jump activity parameters from high-frequency observations of the process and
prove that the proposed estimators are rate optimal up to a logarithmic factor.
