In the case of orientable closed $3$-manifolds, we have 4 ingredients that ensure parallelizability:
1a) Closed smooth $n$-manifolds have homotopy type of a $CW$-complex
1b) Closed smooth $n$-manifolds are Poincare duality spaces.
2) First Stiefel-Whitney class is $0$, from orientability
3) $BSU(2)$ is $3$-connected
1b) gives us Poincare duality and all of the familiar machinery of Steenrod squares and Wu's classes, which, combined with 2), provides for the vanishing of higher classes.
This, in turn, allows us to construct a map to $BSU(2)$. Throwing in 1a), we have that this map is null-homotopic QED.
For open 3-manifolds, I don't see why the computations with characteristic classes should still be valid (as 1b doesn't hold), but one can still show that the map to $BSU(2)$ is null-homotopic, provided it exists.
I am aware of Whitehead's paper https://www.sciencedirect.com/science/article/pii/B9780080098722500260, but I haven't read it, and I don't know if he uses obstruction theory, or something else.
Can the classical proof using obstruction theory still be salvaged in open case?
EDIT: Per Andy Putman's comment, I've replaced the false statement in 1b).