I know "parts of speech" is used to mean nouns, adjectives, verbs, etc., but is there a word or phrase for "types of nouns" that would include subject, direct object, indirect object, etc.?
There are various terms for the parts of a sentence (Oxford Guide to English Grammar: subject, verb, object, complement, adverbial): sentence parts, sentence elements, constituents.
By the way "verb" is a word class, so it is no good and precise term for the SE (sentence element). Verb part or verb element would be better. Likewise, adverbial is used as word class for adverb groups. So, in order to have a precise term for the SE adverbial part/element would be better.
After some more research, it seems like noun functions is commonly used to include all the things a noun can be, including subject, direct object, indirect object, prepositional object, etc.