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We study the problem of function inversion with preprocessing where, given a function f : [N] → [N] and a point y in its image, the goal is to find an x such that f(x) = y using at most T oracle queries to f and S bits of preprocessed advice that depend on f. The seminal work of Corrigan-Gibbs and Kogan [TCC 2019] initiated a line of research that shows many exciting connections between the non-adaptive setting of this problem and other areas of theoretical computer science. Specifically, they introduced a very weak class of algorithms (strongly non-adaptive) where the points queried by the oracle depend only on the inversion point y, and are independent of the answers to the previous queries and the S bits of advice. They showed that proving even mild lower bounds on strongly non-adaptive algorithms for function inversion would imply a breakthrough result in circuit complexity. We prove that every strongly non-adaptive algorithm for function inversion (and even for its special case of permutation inversion) must have ST = Ω(N log (N) log (T)). This gives the first improvement to the long-standing lower bound of ST = Ω(N log N) due to Yao [STOC 90]. As a corollary, we conclude the first separation between strongly non-adaptive and adaptive algorithms for permutation inversion, where the adaptive algorithm by Hellman [TOIT 80] achieves the trade-off ST = O(N log N). Additionally, we show equivalence between lower bounds for strongly non-adaptive data structures and the one-way communication complexity of certain partial functions. As an example, we recover our lower bound on function inversion in the communication complexity framework.