  We relate the moduli space of analytic equivalent germs of reduced
quasi-homogeneous functions at $(\mathbb{C}^2,0)$ with their bi-Lipschitz
equivalence classes. We show that any non-degenerate continuous family of
(reduced) quasi-homogeneous functions with constant Henry-Parusi\'nski
invariant is analytically trivial. Further we show that there are only a finite
number of distinct bi-Lipschitz classes among quasi-homogeneous functions with
the same Henry-Parusi\'nski invariant providing a maximum quota for this
number.
