I have found three different words meaning parents-in-law:
But I'm sure the words are not interchangeable.
How are the words different? What would be the right way to use each of the words?
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A corpus is a good tool to answer this type of question yourself.
IMHO, 舅姑 sounds old, and it may have an unwanted connotation (the kanji 姑 is often used to refer to bad MIL/DIL relationships, aka 嫁姑問題). If you want a safe and neutral word, I recommend 義父母 or 義理の親.