Is it permissible to initiate a discussion with the female doctor about marrying her daughter and planning the engagement, or should the conversation with her or her husband be in the presence of the parents, knowing that the questioner is certain of his parents' approval?

You are permitted to speak with the woman regarding the proposal for her daughter, as needed, while observing the Shariah-compliant guidelines and avoiding temptation. It is preferable to propose to her guardian, especially since some scholars believe the guardian has the right to refuse the proposal if the girl is forced. A forced girl is one whose father compels her to marry against her will. The presence of your parents is not a condition, but if possible, their presence is more appropriate.