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We study Yang-Baxter equations with orthosymplectic supersymmetry. We extend a new approach of the construction of the spinor and metaplectic Rˆ-operators with orthogonal and symplectic symmetries to the supersymmetric case of orthosymplectic symmetry. In this approach the orthosymplectic Rˆ-operator is given by the ratio of two operator valued Euler Gamma-functions. We illustrate this approach by calculating such Rˆ operators in explicit form for special cases of the osp(n|2m) algebra, in particular for a few low-rank cases. We also propose a novel, simpler and more elegant, derivation of the Shankar-Witten type formula for the osp invariant Rˆ-operator and demonstrate the equivalence of the previous approach to the new one in the general case of the Rˆ-operator invariant under the action of the osp(n|2m) algebra.