The way I think of the relationship of puzzles to story is best illustrated by an example from Wodehouse. Spoilers for Jeeves and the Yule-tide Spirit follow:
In the short story Jeeves and the Yule-tide Spirit, Bertie Wooster has fallen head-over-heels in love with Bobbie Wickham, a mischievous and somewhat high-spirited young lady. Bobbie suggests that Bertie plays a trick on Tuppy Glossop, a mutual friend, and eager to impress her, Bertie agrees.
The trick involves sneaking into Tuppy’s bedroom in the middle of the night and puncturing his hot water bottle with a darning needle tied to a long stick. Unfortunately Tuppy has swapped rooms with Sir Roderick Glossop and all manner of chaos results.
But here’s the thing - obtaining some thread, a darning needle and a long stick, tying one to the other, obtaining access to a room and puncturing a hot water bottle are absolutely the stuff of text adventure puzzles. But it’s the motivation behind these actions that ties the puzzle to the story - Bertie’s not doing all this stuff arbitrarily like the protagonist of Zork, he’s doing it because he wants to walk down the aisle with Bobbie Wickham.
The parser game is very good at modelling physical objects, and very bad at modelling things like love and deceit and persuasion, but it’s easy enough to tie one to the other.
“You mean to say that, after she had put me up to the scheme of puncturing Tuppy’s hot-water bottle, she went away and tipped Tuppy off to puncturing mine?”
“Precisely, sir. She is a young lady with a keen sense of humour, sir.”
I sat there, you might say stunned.
– Jeeves and the Yule-tide Spirit by P. G. Wodehouse