Cold molecules provide an excellent platform for quantum information, cold chemistry, and precision measurement. Certain molecules have enhanced sensitivity to beyond standard model physics, such as the electron's electric dipole moment (eEDM). Molecular ions are easily trappable and are therefore particularly attractive for precision measurements where sensitivity scales with interrogation time. Here, we demonstrate a spin precession measurement with second-scale coherence at the quantum projection noise (QPN) limit with hundreds of trapped molecular ions, chosen for their sensitivity to the eEDM rather than their amenability to state control and readout. Orientation-resolved resonant photodissociation allows us to simultaneously measure two quantum states with opposite eEDM sensitivity, reaching the QPN limit and fully exploiting the high count rate and long coherence.