  We investigate using optically trapped linear polyatomic molecules as probes
of nuclear spin-dependent parity violation. The presence of closely spaced,
opposite-parity $\ell$-doublets is a general feature of such molecules,
allowing parity-violation-sensitive pairs of levels to be brought to degeneracy
in magnetic fields typically 100 times smaller than in diatomics. Assuming
laser cooling and trapping of polyatomics at the current state-of-the-art for
diatomics, we expect to measure nuclear spin-dependent parity-violating matrix
elements $iW$ with 70 times better sensitivity than the current best
measurements. Our scheme should allow for 10 \% measurements of $iW$ in nuclei
as light as Be or as heavy as Yb, with averaging times on order the of 10 days
and 1 second, respectively.
