Moffit and Butera explain the overview of the debate:
pg. 166 "Second, in most of its occurrences, the translators of the LXX rendered [ger] with one of two words: [proselutos] or [paroikos]. Of these, they preferred [proselutos] in the vast majority of cases. While Geiger took this to imply that the Greek terms were synonymous, Allen claimed, '[T]he [LXX] version itself, when carefully examined, tells a very different tale.' As noted above, passages such as...Deut 14:21 use [paroikos] to render [ger]. According to Allen, [ger] in such passages 'cannot mean a proselyte, but must denote members of a tribe or nation sojourning in a strange land.'" (pg. 166 of "New Evidence for the Meaning and Provenance of the Word [Proselutos]" by Moffit and Butera.
pg. 174 "Allen's appeal to Exod 12:48-49 appears, therefore, to have begged the question. The claim that the translators took the [ger] to be a convert and used [proselutos] to indicate that interpretation is not proven by pointing to a passage such as Exod 12:48-49 unless one presupposes that [proselutos] means proselyte. As it stands, the Greek text, and particularly the conditional construction, effectively captures the biblical meaning of [ger]--the resident alien must be circumcised if he wants to keep Passover. Thus, these verses actually refute Allen's thesis and confirm Geiger's position--[paroikos] and [proselutos] appear to be roughly synonymous for the translators of the LXX." pg. 174 of "New Evidence for the Meaning and Provenance of the Word [Proselutos]" by Moffit and Butera.
Torrey Seland in "[Paroikos Kai Parepidemos]: Proselyte Characterizations in 1 Peter?", disagrees with Allen as well and states, '[paroikos]...[is a] proseltye-related [term]," pg. 252.
Hegg seems opposed to the idea of paroikos and proselutos being synonymous:
"The LXX translators apparently recognized this distinction, for in Deuteronomy 14:21 they translate 'alien who is in your town' [to 'paroikos']...while Leviticus 17:15 employs [proselutos] to translate ger. I might suggest that in this case [paroikos] is a more general term, and that [proselutos] a more specific term, denoting one who had become a part of the clan."
I'd like to provide a little more ammunition to Hegg's assertion.
Novak examines De Vaux who explains, in light of the paroikoi of the Ptolemaic empire or the perioikoi of Sparta, the term paroikos denotes those aliens who have no political rights:
"De Vaux later compares the gerim with the perioikoi of ancient Sparta, the original inhabitants of the Peloponnese who retained their freedom but had no political rights....Throughout the ancient world, where full citizenship was determined by patrimony, provision had to be made for resident aliens, for it was neither possible nor practical to enslave them all. In Athens such resident aliens were called metoikoi; in Sparta, as we have seen, perioikoi. In the Ptolemaic empire they were designated paroikoi or katoikoi. All of these terms derive from the Greek oikiein, 'to dwell,' just as the term ger comes from the Hebrew gur, having the same meaning...This parallelism was clearly recognized by the ancient translations of the Bible, most notably the Septuagint and the Vulgate...the term proselyte is a neologism of the Septuagint based on proselthein, 'to come to.' It suggests a religious newcomer as opposed to the older epelytos, which had a more secular meaning," pg. 22 of Novak's "The Image of the Non-Jew in Judaism"
We also see this understanding of paroikos as being "an alien without political rights" in the Apostolic writings. In Ephesians 2, Paul contrasts the Gentile's former status as "paroikos" with his new status as "politeia", a term that Paul used elsewhere to refer to his own Roman citizenship.
Also, Hegg is write to say that the proselyte is one who has become family. This is evident from the cognate of proselutos, "proserchomai" which means one who is brought near. The rabbis and also Paul talk about the Gentiles being brought near via blood--with Paul citing to Yeshua's blood. And blood is family.
Now the translators of the LXX were scribes and were thus authoritative interpreters of the meaning behind the MT. They also were familiar with the etymology and sociological history of terms such as paroikos. So I agree with Hegg that the terms are not synonymous.
Gotta run.
Shalom,
Peter