What are the differences between these three words linguistically?

Question

According to Annals of Tacitus there are three words that derive from "Arsac":

• Arsacis (e.g. VI:33)
• Arsacidis (XIII:37)
• Arsacidarum (e.g. XII:10)

What are the differences between these three words in Latin grammar?

Answer 1

Tacitus gets his names from Greek:

• Greek Ἀρσάκης became Latin Arsacēs, i.e. Arsaces, the Parthian king,
• Greek Ἀρσακίδης became Latin Arsacida, i.e. Arsacid, one of his dynastic successors.

Arsacis is a genitive singular of the first word: "of Arsaces". This treats the word as if it's in the third declension with a stem Arsac-, which is unusual (the Greek is in the first declension) but not extremely so.

Arsacidis is a dative plural (formally it could also be an ablative plural, but in this passage it's a dative) of the second: "to the Arsacids". Arsacidarum is a genitive plural of the same: "of the Arsacids". Here Tacitus treats the word as being in the first declension, just as the Greek is, even though masculine first-declension words are comparatively very rare in Latin.

The reason the first word gets put in the third declension is because personal names tend not to get changed more than is necessary, which leads to awkward fits in the grammatical system, and unless a noun is formally a very obvious match for the first or second declension (i.e. the nominative ends in -a, -us, or -um) it defaults to the third. (If Ἀρσάκης were adapted to the first declension it would be Arsaca.)
The second word does get adapted to the Latin first declension because people did realise the Greek first declension was equivalent to the Latin first declension in some way, and as it's not a person's name, more leeway was afforded to change it.