  We obtain asymptotics of polynomials satisfying the orthogonality relations
$$ \int_{\mathbb{R}} z^k P_n(z; t , N) \mathrm{e}^{-N \left(\frac{1}{4}z^4 +
\frac{t}{2}z^2 \right)} \mathrm{d} z = 0 \quad \text{ for } \quad k = 0, 1,
..., n-1, $$ where the complex parameter $t$ is in the so-called two-cut
region. As an application, we deduce asymptotic formulas for certain families
of solutions of Painlev\'e-IV which are indexed by a non-negative integer and
can be written in terms of parabolic cylinder functions. The proofs are based
on the characterization of orthogonal polynomials in terms of a Riemann-Hilbert
problem and the Deift-Zhou non-linear steepest descent method.
