You are describing a preference between two choices.
1) go to a party with people you can't stand.
2) upset your friend.
You use "would rather" and "than" to indicate your preference.
Your sentence structure is this: I would rather ____(preferred choice)____ than _____ (not the preferred choice).
Therefore, your sentence should be: I would rather go to parties with people I can't stand than upset my friend.
If you use "instead of", it describes an unexpected outcome.
Example: instead of showing appreciation, he walked away. (Unexpected)
Your sentence is describing things that you do but you prefer not to do. Therefore, "instead of" is not appropriate in this case.
Hope this helps.