We present a general approach for calculating new electron-electron interactions with atomic structure theory, include dominant many-body effects, and extract bounds for parity-odd cases. The procedure is valid for any mediator mass and improves previous calculations by including many-body corrections to the new interaction through the time-dependent Hartree-Fock method. Bounds on coupling strengths are then extracted for scalar-pseudoscalar and vector-axial vector interactions by comparison with experimental atomic electric dipole moments and parity non-conserving (PNC) transitions respectively. In the latter case, our calculations rule out any coupling-mass ratio $|g_V^e{g}_A^e|/m_X^2$ greater than ${10}^{-11}Me{V}^{-2}$ above $m_X\sim10\,{\rm MeV}$, covering a previously unconstrained mass region. We also report an updated calculation of the regular (Standard Model) electron-electron $Z^0$ exchange contribution to the $6s$-$7s$ PNC transition amplitude $E_{PNC}$ in cesium.