Motivated by the challenges and limitation of existing experimental techniques, this work elaborates our recent proposal for the precision measurement of atomic parity nonconservation by quantum metrology with an entangled pair of trapped ions. Although the technique is fundamentally based on an existing proposal, it can potentially reach a figure of merit twice as high for an exceedingly tiny observable, yet to be measured in any ion trap experiment. The systematic of the technique have been shown to be better controlled as compared to other experiments. Since the goal of the experiment is to test the Standard Model of physics, the measured parity violating transitional dipole matrix element has to be compared with the theory of competitive precision. The choice of the element therefore plays an important role for the success of the experiment. Barium and radium ions have been compared from this perspective. The work presents the theoretical study on nuclear spin dependent atomic parity nonconservation and nuclear anapole moment in different isotopes of these ions and proposes specific transitions that are experimentally feasible.