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I am not aware of a single Fenchel-Nielsen type parameterization as you ask for, and I'm not sure there can be one, because even on the boundary sphere the manner in which Fenchel-Nielsen coordinates are "re-adapted" does not produce a single coordinate system. The way that proof gets coordinates for the boundary sphere does start with a pants decomposition of the surface, as do Fenchel-Nielsen coordinates. And one does get a single coordinate system for the open subset of the boundary sphere which has nontrivial intersection number with each pants curve. But then one has to patch in additional coordinate charts to cover the closed subset of measure foliations that have zero intersection number with one or more pants curve. The proof does demonstrate that these coordinates can be patched together in such an explicit way that one can see the homeomorphism to a sphere, but nonetheless one is still patching things up.

Edit: I also recall that in one of his very earliest writings on this topic, Thurston gave a different proof, certainly not explicit, that the boundary is a sphere. Namely, from the existence of a pseudo-Anosov homeomorphism $\phi$, which acts with attractor--repeller dynamics, one gets a covering by two open sets homeomorphic to Euclidean space: for any neighborhood $U_+$ of the attracting fixed point and any neighborhood $U_-$ of the repelling fixed point there exists $n>0$ such that $\phi^n(U_-)$ and $U_+$ cover the boundary. It follows that the boundary is homeomorphic to a sphere. I posted this question to verify that the same was true for manifolds with boundary, and so the same proof works for compactified Teichmuller space, as Thurston undoubtedly knew: the action of a pseudo-Anosov homeomorphism on compactified Teichmuller space also has attractor-repeller dynamics, and so it is covered by two manifold-with-boundary coordinate charts, and so it is homeomorphic to a closed ball.

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I am not aware of a single Fenchel-Nielsen type parameterization as you ask for, and I'm not sure there can be one, because even on the boundary sphere the manner in which Fenchel-Nielsen coordinates are "re-adapted" does not produce a single coordinate system. The way that proof gets coordinates for the boundary sphere does start with a pants decomposition of the surface, as do Fenchel-Nielsen coordinates. And one does get a single coordinate system for the open subset of the boundary sphere which has nontrivial intersection number with each pants curve. But then one has to patch in additional coordinate charts to cover the closed subset of measure foliations that have zero intersection number with one or more pants curve. The proof does demonstrate that these coordinates can be patched together in such an explicit way that one can see the homeomorphism to a sphere, but nonetheless one is still patching things up.
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