  n this paper we show the existence of traveling waves $w: [0,+\infty) \times
\mathbb{R}^2 \to \mathbb{R}^k$ ($k \geq 2$) for the parabolic Allen-Cahn system
\begin{equation} \partial_t w - \Delta w = -\nabla_u V(w) \mbox{ in }
[0,+\infty) \times \mathbb{R}^2, \end{equation} satisfying some
\textit{heteroclinic} conditions at infinity. The potential $V$ is a
non-negative and smooth multi-well potential, which means that its null set is
finite and contains at least two elements. The traveling wave $w$ propagates
along the horizontal axis according to a speed $c^\star>0$ and a profile
$\mathfrak{U}$. The profile $\mathfrak{U}$ joins as $x_1 \to \pm \infty$ (in a
suitable sense) two locally minimizing 1D heteroclinics which have different
energies and the speed $c^\star$ satisfies certain uniqueness properties. The
proof of variational and, in particular, it requires the assumption of an upper
bound, depending on $V$, on the difference between the energies of the 1D
heteroclinics.
