SortShort answer: Threin
Long answer: It depends...
I just went down a rabbit hole trying to figure this out, and I believe I've found the answer.
Firstly: In my understanding of the modern usage of "twain", it's not fully synonymous with "two". "twain" suggests a connection or similarity between the parts, while "two" carries no such connotation.
Q: Is there a modern equivalent of "twain", but for "three parts"?
A: No
Q: Is there a historic equivalent of "twain", but for "three parts"?
A: Yes!
Middle English
twein/tweine/tweien(e)/tweiin(e)/twen(e)/twain(e)/...
Twain is just one of the fourty-something spellings of the same word, and their use is significantly more varied in Middle English than how twain is used today. It seems to be more commonly used in the same manner as twice, but I believe I've found a couple of instances where we can recognize its use as being pretty close to its modern counterpart:
Cloven in twain:
a1475 Liber Cocorum (Sln 1986)p.43 : Take almondes cloven in twen, Þat fryid ar with oyle, and set with wyn..And serve hit forthe.
Divided into twain:
a1450 Aelred Inst.(2) (Bod 423)16/621 : Charite is dyuyded in-to tweyne, loue of God and loue of thy neighboure.
Source:
The Middle English Compendium: Twein - see definition 2. (a)
thrin/thrinne/thrine/thriin/threin
Threin sees much less overall use, but I believe the first two of the following instances contain the usage we're looking for. I struggle with interpreting the third, but felt it might be relevant. And no, it does not originate from Latin, but rather from the Late Old English þrinna.
Burst in threin:
a1400(a1325) Cursor (Vsp A.3)9815 : His hert aght ar at brest in thrin Ar fra his comamentes tuin.
In threin his goods did he split:
a1400(a1325) Cursor (Vsp A.3)10178 : In thrin his godes did he dele.
(Uncertain)
a1450(a1338) Mannyng Chron.Pt.1 (Lamb 131)385 : Of Brutus cam sire Lokeryn..Iulyn: How þey departed þys land in þrynne, Þat may ȝe heren wel wyþynne.
Source:
The Middle English Compendium: Thrin - see definition 1. (b)
Conclusion
Threin is similar to twain in its form and feel; they share the same origin; and was used roughly to mean the same thing.
I.e.: Threin is to three, what twain is to two.
Disclaimer: This isn't my field of expertise. I'm just a fan of linguistic intricacies, and this answer is the result of only a single evening's research. I'll try to be prompt in fixing mistakes, but please check the comments for corrections.