Oooh tricky. Since you are refering to two people this is an apostrophe after a plural. So the correct spelling is parents'. Regardless of their spacial relationship you are still refering to both parents. If you just say parent's you negated the plurality of the statement and you could only be refering to one parent. Since parent is a singular noun it still follows normal apostrophe rules. If you are this advanced in English writing you may want to purchase the book, "Elements of Style," (title of books should be in italics but I don't have that option) by William Strunk and E. B. White.

Well, it depends. If you're talking about TWO OR MORE parents, then you go with "parents' shoulders", but if you're talking about only one person, you should use " parent's shoulders", thought it'd be more common to be used "mother" or "father" instead of "parent" when it's only one person.