Parallelisable three dimensional orientable manifolds and Stiefel's theorem
Stiefel's article deals only with closed (= compact, without boundary) manifolds. The first paragraph of it says:
Die $n$-dimensionalen Mannigfaltigkeiten, welche in dieser Arbeit betrachtet werden, sollen geschlossen und stetig differenzierbar sein.
which roughly means:
The $n$-dimensional manifolds which will be considered in this work will be closed and continuously differentiable.
The theorem is nonetheless true for the other orientable $3$-manifolds as well. Actually, we have a stronger result.
Theorem (Whitehead, 1961). Let $M$ be a non-closed connected orientable $3$ manifold. Then there is an immersion $M \to \mathbb R^3$.
The reference is Whitehead, The Immersion of an open 3-Manifold in Euclidean 3-space, Proc. London Math. Soc., 1961. The proof uses very different techniques than the standard one for the closed cases (the ones quoted in the question you mentioned). By puncturing a closed $3$-manifold, it is actually easy to deduce the closed case from the non-closed one.
Remarks on Kirby's proof. In Kirby's little The Topology of 4-Manifolds (which features a very small but quite brilliant chapter on 3-manifolds), one can find some remarks proving that the theorem for the closed case actually prove the general theorem. Let me summarise these remarks.
First, the "closed" theorem implies the "compact with boundary" (or "bounded" in a quite old but charming terminology) theorem: if $M$ is a compact manifold, its double $DM$ is a closed one, and it is oriented if $M$ is oriented. The triviality of $T_{DM}$ implies the triviality of its restriction $T_{M} = \left(T_{DM} \right)_{|M}$.
Then, Kirby claims without proof that if a non-compact manifold $M$ has a non-trivial tangent bundle $T_M$, then the tangent bundle is already non-trivial in restriction to a "compact piece". Since every compact part of $M$ is included in a compact codimension-0 submanifold $N$, the previous step shows that such a situation is impossible.
I actually believe Kirby's claim, but I was unable to prove it. It turns out that more general statements are actually false, which surprised me a little. This proof and the subsequent question on the validity of what I've called Kirby's claim have already come up on MSE, but I haven't been able to find a definitive answer. Maybe it would be worth another question.