Each blank in this sentence needs its own type of preposition: one for a location, one for showing which thing belongs to which, and one for a stretch of time. Let's test each option in the full sentence.
- at; of; for: "parted at the door of the cabin that I had rented for the night" reads naturally, "at" marks the doorway where they separated, "of" links the door to the cabin, and "for" gives the length of time the cabin was rented.
- for; at; of: this gives "parted for the door", which does not make sense, since "for" cannot mark a place of separation.
- of; for; in: this gives "parted of the door", again wrong, since "of" does not fit after "parted" to show a location.
- in; of; for: this gives "parted in the door", an odd phrase, since people part "at" a doorway, not "in" it.
Only the first option keeps every blank grammatically natural, so "at; of; for" is correct.
Let's summarize:
- "At" fits a point of location, "of" fits a belonging or association, "for" fits a duration.
- Testing each option inside the full sentence quickly rules out the other three.
So the correct sequence is "at; of; for".