  For the planar $N$-centre problem $$ \ddot x = - \sum_{i=1}^N \frac{m_i
(x-c_i)}{| x - c_i|^{\alpha+2}}, \qquad x \in \mathbb{R}^2 \setminus \{
c_1,\ldots,c_N \}, $$ where $m_i > 0$ for $i=1,\ldots,N$ and $\alpha \in
[1,2)$, we prove the existence of entire parabolic trajectories, having
prescribed asymptotic directions for $t \to \pm\infty$ and prescribed
topological characterization with respect to the set of the centres.
