"My parents' friendship with..." vs. "my parent's friendship with..." vs. "my parents friendship with..."

Solution 1:

"My parents' friendship with Sally's parents" is correct.

When you're confused about apostrophe usage, try substituting the word "of" into the sentence to see if it makes sense.

"The friendship of my parents with the parents of Sally" is grammatically clear (if stilted and certainly not recommended, in speech or in writing).

Semantically,

the friendship of my parents = my parents' friendship

and

the parents of Sally = Sally's parents

Thus, in the actual sentence, you would need an apostrophe after both "my parents" and "Sally".

Solution 2:

The first is correct if you are referring to both of your parents. The second is correct if you only referring to one parent. I don't think the third is correct in any situation.

Solution 3:

Apostrophes for ownership can be tricky, but here's a strategy for you.
1. Figure out who the owner is.
2. Put the apostrophe immediately after the owner.
3. Add "s" if needed.

If the friendship "belongs" to both of your parents: The parents have the friendship, so we immediately follow parents with the apostrophe. Then we add "s" as needed. (Here, we don't need the final "s" because parents is plural noun that ends in "s.") The result is parents'.

If the friendship "belongs" to one parent: The parent has the friendship, so we immediately follow parent with the apostrophe. The we add "s" as needed. The result is parent's.