  Let $G$ be a reductive group. We prove that a family of polynomial actions of
$G$ on $\mathbb{C}^2$, holomorphically parametrized by an open Riemann surface,
is linearizable. As an application, we show that a particular class of
reductive group actions on $\mathbb{C}^3$ is linearizable. The main step of our
proof is to establish a certain restrictive Oka property for groups of
equivariant algebraic automorphisms of $\mathbb{C}^2$.
