Hi Ya Wen,
To answer your first question: Yes, you can certainly say "Can you find the different parts OF the story?". Here, you are using the preposition "of" to express a part-whole relationship. For example, a story is presented to you. You read the story and sieve out the parts which are different from one another.
However, if you say "Can you find the different parts TO the story?", there may be a slight change in focus. You could be piecing together parts which are different to form an entire story.
The difference is very slight and most people would still understand what you mean if you use either.
Just another example to illustrate what I mean:
"These are two separate clues TO the answer." (These clues LEAD YOU to the answer.)
"These are two separate clues OF the answer." (Sounds odd. These clues are not PART of the answer. They are merely providing hints to the answer.)
"These are two separate sections of the answer." (Each section is a PART of the answer.)
Now, to your second question:
1) You can also use "different" with "from". In this case, we are focusing on the dissimilarity of something from the rest. Example: This essay is very different FROM his previous ones. / My experience tonight in this restaurant is different FROM my previous ones.
2) You can also use "as different as" in a simile to compare the qualities of two different things. Example: The sisters are AS different AS day and night.
3) You can also use "different" with "by". Here, we are focusing on the degree of difference. Example: Genetically, chimps and humans are different BY just 1% of our DNA.
I hope this helps you.