I don't really know about the state of the art, but I've come across a couple of examples in the literature at least. In the article:

Frank, David L., *An invariant for almost-closed manifolds*, Bull. Amer. Math. Soc. 74 (1968) 562–567, MR0222906

the unique exotic $8$-sphere and an order-$3$ element of $\Theta_{10} \cong \mathbb{Z}/6$ are shown to be in the image of Milnor's plumbing construction; see Examples 1 and 2 on page 565. Since they're exotic (i.e. not the standard sphere) and even-dimensional, they do not bound any parallelisable manifold (since $bP_{2n+1} = \{S^{2n}\}$). Also, in the article:

Sperança, L. D., *Pulling back the Gromoll-Meyer construction and models of exotic spheres*, Proc. Amer. Math. Soc. 144 (2016), no. 7, 3181–3196, MR3487247

there is an explicit description of clutching diffeomorphisms that realise these two examples as twisted spheres; see Theorem 4.6 (plus the description in the middle of page 3187 of "reentrance").