  We consider and completely solve the parametrized family of Thue equations
\begin{eqnarray*}X(X-Y)(X+Y)(X-\lambda Y)+Y^4=\xi,\end{eqnarray*} where the
solutions $x,y$ come from the ring $\mathbb{C}[T]$, the parameter
$\lambda\in\mathbb{C}[T]$ is some non-constant polynomial and
$0\neq\xi\in\mathbb{C}$. It is a function field analogue of the family solved
by Mignotte, Peth\H{o} and Roth in the integer case. A feature of our proof is
that we avoid the use of height bounds by considering a smaller relevant ring
for which we can determine the units more easily. Because of this, the proof is
short and the arguments are very elementary (in particular compared to previous
results on parametrized Thue equations over function fields).
