Mathieu Baillif provided an answer to the question. I'll leave here some more notes for future readers. The following strengthening of the theorem in Gauld's book "Non-metrizable manifolds" holds:
Let $X$ be a Hausdorff space, and $X' \subset X$. Then $X'$ is collared in $X$ and $X'$ is strongly paracompact in $X'$ if and only if $X'$ is closed in $X$, $X'$ is strongly paracompact in $X$, and $X'$ is locally collared in $X$.
Suppose then that $X$ is a manifold-with-boundary; e.g Hausdorff and locally upper-Euclidean and not necessarily connected. It can be shown that $\partial X$ is closed in $X$, and $\partial X$ is locally collared in $X$. It can also be shown that in a locally compact space, which is a manifold-with-boundary is, strong paracompactness is equivalent to paracompactness. Therefore, we have the following theorem:
Let $X$ be a manifold-with-boundary. Then $\partial X$ is collared in $X$ and $\partial X$ is paracompact in $\partial X$ if and only if $\partial X$ is paracompact in $X$.
This is quite a nice theorem. However, it leaves open the characterization of of non-paracompact collared boundaries. I would be interested to hear if someone can complete the characterization.
The following example shows that there exists non-paracompact collared boundaries. The open long ray is a non-paracompact manifold. Therefore, its cartesian product with the interval $[0, 1)$ is a manifold-with-boundary whose boundary is non-paracompact and clearly collared.