A supersymmetric theory with osp(1|2)-invariance can be used to describe spanning (hyper-)forests on a given graph. In a sigma-model, the bosonic part of the superfield can be integrated out and the symmetry is realized non-linearly in the remaining Grassmann degrees of freedom. We investigate the invariant osp(1|2) Grassmann and Clifford subalgebras. We discover that, given an arbitrary vertex ordering, a basis of independent elements is given by terms labeled by non-crossing partitions, respectively of the vertices of the graph, and of a doubling of this set. The invariant Clifford subalgebra is isomorphic to a Temperley--Lieb Algebra, in a peculiar even/odd generalization. A relation involving four complex fermions plays a central role in the construction, and disentangles the only 4-set crossing partition as a linear combination of the other partitions with up to two blocks. This relation appears in different flavours, and equivalence of the formulations is shown.