Given a pair of finite posets $A \subseteq P$, the function counting integer-valued order preserving extensions of an order preserving map $λ: A\rightarrow \mathbb{Z}$ from $A$ to $P$ is given by a piecewise polynomial in $λ$. We provide a criterion for the nonnegativity of the coefficients of these multivariate polynomials and apply it to show that marked order polytopes of skew shapes are Ehrhart positive in a multivariate sense. This extends recent results of Ferroni-Morales-Panova on order polytopes of skew shapes and proves conjectures on the Ehrhart positivity of skew Gelfand-Tsetlin polytopes and $m$-generalized Pitman-Stanley polytopes due to Alexanderson-Alhajjar and Dugan-Hegarty-Morales-Raymond, respectively.