You can, but strictly speaking its position now at the back is better.

Yes

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

我有一年没见过我父母了。 I haven't seen my parents for a year.

我去年没见过我父母。 I didn't saw my parents last year.

了is like have/has/had in perfect tense.

I have the same question. I tried "我沒有見父母一年，我想他們。", which was rejected.

Also: Oddly, the suggested answer, "我沒有見父母一年了，我想他們。", had the comma underlined instead of the "了".