Antonyms for the phrase "greater parts" are "smaller portions," "lesser segments," or "minor parts." These words suggest a smaller size or quantity as opposed to a larger one. When we talk about greater parts, we often refer to a significant portion or the majority of something, while antonyms emphasize the opposite, representing only a smaller portion or a minority. For instance, when dividing a cake among five people, a greater part could be three pieces, and the antonym, smaller portions, could be two pieces shared among the remaining two persons. Therefore, using antonyms can be helpful in expressing the opposite meaning to greater parts to better communicate and avoid misunderstandings.