There are two variations on Aristotelian ontology. On the sparser version there are substances and their modes (accidents and essences). On the more bloated version there are substances, modes and (proper) parts. I want to argue that the more bloated version should be reduced to the sparser one.
Parts in an Aristotelian ontology are unlike the parts of typical contemporary ontologies. They are not substances, but rather they are objects that depend on the substance they are parts of. At least normally when a part, say a finger, comes to be detached from the substance it is a part of, it ceases to exist—a detached finger is a finger in name only, as Aristotle insists.
This makes the parts of Aristotelian ontology mode-like in their dependence on the whole. Ockham's razor then suggests that rather than supposing three fundamental categories—substances, mode and parts—we will do better to posit that a part is just a kind of mode. Thus, I really do have a heart, but my heart is just much a mode or accident of me—my cardiacality—as my knowing English is. Both my heart and my knowledge of English confer on me certain causal powers and causal liabilities (knowing English makes me liable to having my feelings hurt by uncomplimentary assertions in English!)
This is not an elimination of parts. Some of my accidents are parts and others are not. Which ones? I do not know. Maybe those accidents that occupy space are parts and those accidents that do not are not. My knowing English doesn't occupy space, while my cardiacality is somewhat vaguely but really located located in space.
Perhaps we need a finer distinction, though. Consider the strength of my arm. This isn't a part of me, but it seems to be located in my arm. I suggest that we distinguish between three ways that a mode can get a location. It can (a) inherit a location from a subject, or (b) it can inherit a location from its own modes, or (c) it can be located in its own right. I suggest that a mode is a part if and only if it has a location of type (b) or (c). The strength of my arm inherits its location from its subject—my arm—and hence is not a part. (It's important to the full development of this ontology that modes can nest. Thus, my arm is a mode of me, and the strength of that arm is a mode of this mode. Both I and my arm are subjects of the strength of the arm.)
I think the distinction between type (b) and type (c) parts is worth thinking about. Maybe matter, that mysterious ingredient in Aristotelian ontology, can be identified with type (c) parts?