  Twisted nodal superconductors have been shown to exhibit chiral topological
superconductivity under broken time-reversal symmetry. Here we show how a
time-reversal preserving topological superconductivity can be induced in nodal
triplet superconductor multilayers. For a bilayer system, the application of a
Josephson spin current in triplet superconductors induces a non-zero spin Chern
number per node in momentum space. However, we show that stabilizing a
nontrivial global $\mathbb{Z}_2$ invariant requires an odd number of layers. As
a specific example we consider trilayers with three forms of twist: chiral,
alternating and single-layer. For chiral and single-layer case, we find that a
gap opening in the dispersion leads to a non-trivial $\mathbb{Z}_2$ topological
invariant. For single layer twists, we show how this invariant is non-trivial
when extended to an arbitrary odd number of layers.
