Step 1: Understanding the Question.
We must fill three blanks in the sentence "My friend and I parted __ the door __ the cabin that I had rented __ the night" with the correct prepositions, in order.
Step 2: Key Approach.
Each blank needs a different type of preposition: the first names a specific point or position ("at the door"), the second shows belonging or association ("of the cabin", the door belonging to the cabin), and the third names a stretch of time over which something happened ("for the night", the cabin rented for the whole night).
Step 3: Detailed Explanation.
Blank 1, "parted __ the door": we use "at" with a specific point or location such as a doorway, so "parted at the door" is the natural phrase for where two people separated.
Blank 2, "__ the cabin": we use "of" to show the door belongs to, or is part of, the cabin, so "the door of the cabin" correctly links the door to the cabin it is part of.
Blank 3, "rented __ the night": we use "for" with a duration, since the cabin was rented for the length of the whole night, so "rented for the night" is correct.
This gives the sequence "at; of; for", matching option (A).
Option (B), "for; at; of", would give "parted for the door", which does not make sense, since "for" does not mark a location.
Option (C), "of; for; in", would give "parted of the door", again wrong, since "of" cannot mark where two people separated.
Option (D), "in; of; for", would give "parted in the door", an unnatural phrase, since we part "at" a door, not "in" it.
Step 4: Final Answer.
The correct sequence of prepositions is "at; of; for".
\[ \boxed{\text{at; of; for}} \]