The difficulty is coming up with objective parts.
I think this is made particulary clear when thinking about temporal parts. So you are spread out from birth to death and we can divide you, over time, into parts.
But how could those parts be objective? I might pick 3 second divisons, or 3 year divisions, or daily divisions, but in each case I’ve drawn the lines between the parts.
What are the objective lines between your temporal parts?
Interestingly, this is the same question I asked of David Lewis many years ago. He gave, IMO, the right answer: there are as many divisions among temporal parts as there are among spatial parts. IOW the temporal line is the same as the real number line: it is a continuum with an uncountable infinity of points along it.
Though of course once you get below Planck’s Constant, the divisions no longer make any physical sense.
The more general question about ‘objective parts’ can be answered by reference to ‘natural kinds’. If there are natural kinds in the world, there are objective parts, and our usage of those kind-terms ‘cuts nature at the joints’ (as I think it was Plato who said). If there are no natural kinds, then all divisions of reality into parts are conventional in nature.
But even if such divisions were conventional (e.g., that any such division would be as good, true or useful as any other), that would not change the fact that reality is divisible into parts. The latter is incontrovertibly true.