Why does 了come at the end of the clause here and not directly after the verb? How does that change the meaning?
了 is a very complicated issue. Please check out this website Uses of "le".
But the Chinese translation sounds unnatural anyways, it should be *我一年没有见（我）父母了.
一年 is the predicate here. 一年了 = it has been a year, a year has passed (more literally: it has become a year. this is the same 了 as in 天亮了 "it (the sky) has become bright")