"I have not seen my parents for a year, I miss them."
了 is required in this sentence to express duration of inaction. the form is Subj. + Duration + 没 + Verb + 了. https://resources.allsetlearning.com/chinese/grammar/Expressing_duration_of_inaction
In retrospect: The sentence with "了" is like saying "I haven't seen my parents for a year now." whereas the sentence without "了" is like saying "I didn't see my parents for a year." The former definitely sounds more correct particularly with the added context of "I miss them." Why? Because the first statement (with "了") is explicitly marking the change of state that prompted the second statement.
https://resources.allsetlearning.com/chinese/grammar/Expressing_duration_of_inaction As per above link, the sentence should be "我(已经)一年没有见我父母了，想他们。