When does "part" mean "quarter"?

It's not that the word part has any direct connection with quarter. Tolkien's usage is just a variant on the same trope that gives us...

possession is nine parts/points/tenths of the law

If Tolkien had written "four parts of the folk" (unlikely, I know), it would have meant four fifths. Once you specify how many parts you have, the implication is there's one more unaccounted for.

But I do think three is a good number for these contexts - three quarters is a familiar fraction, two is too small for any expression emphasising scale, and four fifths is getting complicated. We're much more likely to say three parts drunk or three sheets to the wind, rather than four or five.