I will discuss recent joint work with B. Pym where we show that a logarithmic holomorphic Poisson manifold carries a natural mixed Hodge structure on its topological K-theory. In many cases of interest, this mixed Hodge structure detects the parameters appearing in known deformation quantizations of the Poisson manifold. I will demonstrate this phenomenon explicitly for complex Poisson tori, and prove that one recovers the usual 'q-parameters' appearing in the relations of non-commutative tori. Time permitting, I will discuss how cyclic formality relates this Hodge structure to a mixed Hodge structure on the periodic cyclic homology of the quantized algebra.