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 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.

Thoughts on generalization to locally compact space

I think I have proved that in a locally compact Hausdorff space, if a point has a local collar, then it also has a local collar with compact domain. Since a compact Hausdorff space is normal, this takes care of the requirement of normality for the local collar domains in Gauld's proof. Then, instead of requiring strong paracompactness, we may require the existence of local collars whose domains form a star-finite collection. Then the same proof goes through. Hence, we have:

Let $X$ be a locally compact Hausdorff space, and $X' \subset X$. Then $X'$ is collared in $X$ if and only if $X'$ is closed in $X$, and $X'$ is star-finitely locally collared in $X$.

So now the problem reduces to finding a collection of local collars whose domains form a star-finite collection.