Much as is common in how we identify periods with eastern Dyansties, particularly those who make claims to some sort of continuity with previous states under previous dynasties, is to label the state by the ruling dynasty. In this case, it takes its namesake from Arsaces I, the same way that the Achaemenid Empire Persian Empire takes its name from Acahemenes, and the Sassasnid/Sasanian Persian Empire takes its name from Sasan, the father of the first Emperor/King. Realistically, you could call the Parthian Empire the Arascid Persian Empire and people should know what you are talking about; it just so happens that the Parthians, while still Iranian, were not strictly speaking 'Persians' as the later at the time referred to a specific geographic area in the south of Iran.