This cult was rather prominent in 6th and 5th centuries BCE, although Burkert and other historians of religion this version of Apollo was more popular in the Greek East (Asia Minor: around Ephesus, Pergamon, Miletus, etc). It is not uncommon to find it in Greek colony city-states or close to Magna Graecia, of course. The word ouliades, because of the context (Elea) could refer to Parmenides as the native of Elea - which is very common, per example to keep it in context of philosophy; the same way Plato names Iccus as one of the greatest gymnasts the Greek world has ever seen as Iccus of Taranto, an all too common way of referring and naming someone in the ancient and medieval worlds. But Apollo Iētros has an another name, Apollo Oulios, of whom we know was popular among Greeks overall before the dominance of Asclepius which was inaugurated by Hippocrates and the sophists.
With relation to Parmenides, the only known portrait (which isn’t in fact him even though it is attributed to him, but is instead an adaption of Metrodorus of Lampsacus) of Parmenides is a head discovered at Elea in 1966 (probably from a herm) with the inscription ΠΑ[Ρ] ΜΕΝΕΙΔΗΣ ΠΥΡΗΤΟΣ | ΟΥΛΙΑΔΗΣ ΦΥΣΙΚΟΣ [‘Parmenides son of Pyres, Ouliades, natural philosopher’]. The monument itself dates from the first century of the Christian era. I’ve already dwelt on what φυσικός means, and this is corroborated by Theophrastus himself who includes Parmenides in Φυσικῶν δόξαι [‘Opinions of the Natural Philosophers’]. Now, as it concerns Οὐλιάδης, I am now pretty much convinced that what this signifies is the patronymic Οὐλιάδης, namely that it means that the person is ‘’son of Oulis’’. We have numerous examples of this. It recurs at Elea in a fragmentary inscription published in 1970 mentioning Apollo as ἰατρόμαντις [‘physician-prophet’]. There is also another example related to the name Οὖλις [‘Oulis’] in another inscription of the same Julio-Claudian date (1st century AD) from the base of a portrait statue, with the lines inscribed into it that are: Οὖλις Εὐξίνου Ὑελήτης ἰατρὸς φώλαρχος ἔτει τοθ´ [‘Oulis of Hyele, son of Euxinus, physician, pholarch,] in the 379th year’). As A.H. Coxon remarks, there are two similar inscriptions on headless herms, Οὖλις Ἀρίστωνος ἰατρὸς φώλαρχος ἔτει σπ´ ([‘Oulis, son of Aristo, physician, pholarch,] in the 280th year’) and Οὖλις Ἱερωνύμου ἰατρὸςφώλαρχος ἔτει υμς´ ([‘Oulis, son of Hieronymus, physician, pholarch,] in the 446th year’). So these three inscriptions which were found in the same building where evidence related to Parmenides was situated seem to refer to physicians who became officials in a religious association.
Now, these inscriptions purport to be referring to persons officials at least 400 years before their making. A.H. Coxon thinks that because the fragment on Parmenides is the only one discovered in Elea to be undated, he might have been regarded as the patron or founder of the association of Apollo Oulios.
Plutarch (Life of Aristides, 23) points out that Apollo’s name as Οὔλιος means Healer and occurs as a personal name, for instance in Samos in 478: οἱ περὶ τὸν Σάμιον Οὐλιάδην (‘the men of Ouliades of Samos’). The other examples I’ve managed to find and A.H. Coxon mentions in his commentary of Parmenides are Οὐλιάδης Οὐλιάδου [‘Ouliades, the son of Ouliades ’]. Now, can we state with 100% certainty that Parmenides in actuality was a member of such religious association? No, but we have pretty interesting evidence that may point to this fact. The cult of Apollo Oulios was present in Magna Graecia, especially in those colony city-states which were founded by mother city states in Asia Minor. Elea itself was founded in mid 6th century BCE by Phocaea. We know Apollo Oulios was present in Phocaea and Samos, and if we take Parmenides to have been born around 515 BCE by appealing to his visit to Athens in 450 BCE and Plato's description of him as being 65 of age while Zeno was in his early 40s, it does not seem impossible to conclude that there was a sizeable presence of Apollo Oulios in Elea. Pythagorean influence also cannot be ruled out either.
A bit more information on Apollo Oulios. Older sources refer to Oulios as obviously the Healer, which depicts Apollo as the god which is both healer of the gods and brings everything into balance as the measure, but also can very well be someone's undoing as shown in the Iliad and Homeric hymns to Apollo. This paradoxical depiction of Apollo is what is extremely relevant here and belongs to the way in which Greeks understood Apollo historically.