  This work is about parameter estimation for a fast-slow stochastic system
with non-Gaussian $\alpha$-stable L\'evy noise. When the observations are only
available for slow components, a system parameter is estimated and the accuracy
for this estimation is quantified by $p$-moment with $p\in(1, \alpha)$, with
the help of a reduced system through random slow manifold approximation. This
method provides an advantage in computational complexity and cost, due to the
dimension reduction in stochastic systems. To numerically illustrate this
method, and to corroborate that the parameter estimator based on the reduced
slow system is a good approximation for the true parameter value of the
original system, a prototypical example is present.
