We consider the parametric estimation of the volatility and jump activity in a stable Cox-Ingersoll-Ross (α-stable CIR) model driven by a standard Brownian Motion and a non-symmetric stable L ́evy process with jump activity α ∈ (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 [16], 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.