Step 1: Understanding the Concept.
The quickest way to solve this kind of question is to plug each option's three words into the sentence, one full option at a time, and read the whole sentence in your head to see which one sounds like something a native speaker would actually say.
Step 2: Key Formula or Approach.
Prepositions are tied to fixed, memorised collocations rather than logical rules, so the fastest check is: does this exact phrase ('parted at', 'door of', 'rented for') sound like a phrase you have heard before, in that order.
Step 3: Detailed Explanation.
Option (A) gives: 'parted at the door of the cabin that I had rented for the night.' Each piece is a phrase in common use, 'parted at' a place, 'door of' a building, 'rented for' a stretch of time. The whole sentence flows naturally.
Option (B) gives: 'parted for the door at the cabin that I had rented of the night.' 'Parted for the door' and 'rented of the night' are not phrases anyone uses, this sentence sounds wrong throughout.
Option (C) gives: 'parted of the door for the cabin that I had rented in the night.' 'Parted of' is not standard English, and 'rented in the night' changes the meaning to renting while it was night rather than renting for the duration of the night.
Option (D) gives: 'parted in the door of the cabin that I had rented for the night.' The last two blanks, 'of' and 'for', actually work here, but 'parted in the door' is wrong, a person parts at a doorway, not inside it.
Step 4: Final Answer.
Only option (A), 'at; of; for', produces a sentence where every single preposition matches its normal collocation, so that is the correct sequence.