You have to first define what a "part of sentence" is. Your question seems to take part-of-sentence to have to do with combining individual words into constituents, but then you also indicate that part-of-sentence is in part defined by functional role. So an NP would be a part-of-sentence, and a subject, or an agent, or a topic, would be a part-of-sentence. Mixing together all of these notions will naturally lead to incoherence.
Constituent properties have a decent chance for existential universality (that is, there are some that are universal), such as "NP", "Sentence". However, not every theory uses the same labeling (so N" is distinct from NP, and some branches of Minimalism don't bother with labeling which they consider to be epiphenomenal). So you would first have to specify what theory of constituency you assume, if you care about the constituency version of part-of-sentence. (And I suspect that you don't).
The subset of semantic relations that would appear to be parts-of-sentence are probably universal, in that every language has the means of distinguishing such roles (i.e. there are no languages which are incapable of expressing the fact that X is an agent vs. X is a theme; all languages can express the conjunction of two referents). Languages differ wildly in how that distinction is encoded, e.g. by affixation or some kind, by word order, by differential government of agreement, by concatenation with distinct function-marking words.
Questions about grammatical relationship (like subject, direct object) can only be answered in the context of a theory of grammatical relationships. Ergativity is an infamous case of that problem, since one theory might take the surface pattern of affixation or word order to be the primary diagnostic while another would say that an ergative language is nothing more than one where passive is obligatory. And there are theories where relations like "subject", "object" aren't required.