It is NP-hard. The reduction is from $CLIQUE$, so suppose we are given an undirected graph $H$ on $n$ vertices and $m$ edges, with a parameter $k$, and our task is to decide whether $\omega(H)\ge k$. We will need some sufficiently large numbers $M\gg N \gg n$, where we need about $N=n^2$ and $M=n^3$.
The graph $G$ will have two disjoint parts. The first part ...
You may find the following recent reference useful:
M. Caceres, M. Cairo, B. Mumey, R. Rizzi, and A. Tomescu. On the parameterized complexity of the Minimum Path Cover problem in DAGs.
arXiv preprint arXiv:2007.07575