  We prove a Liouville-type theorem for semilinear parabolic systems of the
form $${\partial_t u_i}-\Delta u_i =\sum_{j=1}^{m}\beta_{ij} u_i^ru_j^{r+1},
\quad i=1,2,...,m$$ in the whole space ${\mathbb R}^N\times {\mathbb R}$. Very
recently, Quittner [{\em Math. Ann.}, DOI 10.1007/s00208-015-1219-7 (2015)] has
established an optimal result for $m=2$ in dimension $N\leq 2$, and partial
results in higher dimensions in the range $p< N/(N-2)$. By nontrivial
modifications of the techniques of Gidas and Spruck and of Bidaut-V\'eron, we
partially improve the results of Quittner in dimensions $N\geq 3$. In
particular, our results solve the important case of the parabolic
Gross-Pitaevskii system -- i.e. the cubic case $r=1$ -- in space dimension
$N=3$, for any symmetric $(m,m)$-matrix $(\beta_{ij})$ with nonnegative
entries, positive on the diagonal. By moving plane and monotonicity arguments,
that we actually develop for more general cooperative systems, we then deduce a
Liouville-type theorem in the half-space ${\mathbb R}^N_+\times {\mathbb R}$.
As applications, we give results on universal singularity estimates, universal
bounds for global solutions, and blow-up rate estimates for the corresponding
initial value problem.
